The more fundamental way: Islam is like a math book, you cannot open it up in the page 345, read a theory with all details of its proof and understand it unless you have read its prerequisites in the first 344 pages of the book and even perhaps other books cited therein. There is a proof for the theorem but the reader would not be able to understand it (even may call the proof as misleading and fake!) unless he is qualified to do so. Therefore, like Noah correctly stated you may first try to show him how to crawl before to learn him how to walk and run. Yes, Imam Mahdi AS himself is fundamental to Shi'ite Islam but not more than is The holy Prophet SA and Quran. If he could accept Quran (which is the miracle of the Prophet SA and, thus, would be a firm reason for itself) then you can simply bring on the desk what Quran says about Noah --peace be upon him-- who was a prophet for 950 years (that is, even living more):
وَلَقَد أَرسَلنا نوحًا إِلىٰ قَومِهِ فَلَبِثَ فيهِم أَلفَ سَنَةٍ إِلّا
خَمسينَ عامًا فَأَخَذَهُمُ الطّوفانُ وَهُم ظـٰلِمونَ
And We certainly sent Noah to his people, and he remained among them a
thousand years minus fifty years, and the flood seized them while they
were wrongdoers. [29:14]
Actually Noah's --peace be upon him-- long life is a traditional example from Quran, but maybe a more useful but less noticed example is what Allah states about Yunus --peace be upon him-- after he was swallowed by a big fish: [37:139,140,141,142,143,144]
So also was Jonah among those sent (by Us).
When he ran away (like a slave from captivity) to the ship (fully)
He (agreed to) cast lots, and he was condemned:
Then the big Fish did swallow him, and he had done acts worthy of
Had it not been that he (repented and) glorified Allah,
He would certainly have remained inside the Fish till the Day of Resurrection.
And this means perhaps hundreds of thousands of years remaining alive in such a difficult situation. Also according to Ahadith there we have Khizr, peace be upon him, as one good example who will be alive until the Day of Resurrection as he has drunk from the fountain of life (عین الحیات). Also Allah Azza-va-Jalla says:
وَلَتَجِدَنَّهُم أَحرَصَ النّاسِ عَلىٰ حَيوٰةٍ وَمِنَ الَّذينَ
أَشرَكوا ۚ يَوَدُّ أَحَدُهُم لَو يُعَمَّرُ أَلفَ سَنَةٍ وَما هُوَ
بِمُزَحزِحِهِ مِنَ العَذابِ أَن يُعَمَّرَ ۗ وَاللَّـهُ بَصيرٌ بِما
And you will surely find them the most greedy of people for life -
[even] more than those who associate others with Allah. One of them
wishes that he could be granted life a thousand years, but it would
not remove him in the least from the [coming] punishment that he
should be granted life. And Allah is Seeing of what they do. [2:96]
And by this verse Allah never rejects the possibility for existence of such long lives but only states even if they are granted with such long lives (so it is not impossible) they will anyway die when its time eventually comes and taste the punishment for all their wrongdoings. Also all Muslims according to Quran believe in an eternal life with the same human bodies as our present human bodies after the Resurrection.
After believing in Quran he will certainly believe in the holy prophet --peace be upon him-- the proof of whose prophecy is Quran the miracle. Then he can easily find a bunch of Ahadith which address the issue of Mahdi --peace be upon him--. If he accepted the Hadith of Pond, he will then be a Shia Muslim and then there he can find lots of authentic Ahadith from the holy prophet and his succeeders, Imams, --peace be upon them all-- among which there are witnesses about the very long life of Imam Mahdi, the 12th and last Imam in the Shi'ite school of thought.
The less fundamental but more direct way. People usually and naturally guard against anything that would appear odd with respect to their instantaneous collection of "knowledge and beliefs", so that one should first try to show them why something can be true although seems odd at the first sight, then they would put aside their guard and search to understand the truth, if he care to know the truth indeed. Also there are times that people are taught something and they have trusted their teacher (be it their parents, their close friends, their tutors in university, the distinguished scholars or scientists and etc.) and learnt what they have been taught. Then through the course of time a chain of evidences come to their mentions, no matter however they try not to change their mind they cannot withstand the train, and one day they rebel against all such beliefs that they previously had and anyone who resembles their previous teachers! These people also have produced a natural guard over their mind not to let anyone easily push an idea to their mind. However, if one study this group of people more thoroughly will find out that such persons still quite easily accept data from some new teachers that they think they can now trust in them, usually scientists and philosophers, those who have once set them free from their previous thought cages. Anyway, all of those who have a guard against listening to your new arguments will try not to undergo accepting any new difficulty (duty and etc.) to be introduced to their own life styles unless there is a strong proof for that, as Allah states in [98:1]:
بِسمِ اللَّـهِ الرَّحمـٰنِ الرَّحيمِ، لَم يَكُنِ الَّذينَ كَفَروا مِن
أَهلِ الكِتـٰبِ وَالمُشرِكينَ مُنفَكّينَ حَتّىٰ تَأتِيَهُمُ
Those who disbelieved among the People of the Scripture and the
polytheists were not to be parted [from misbelief] until there came to
them clear evidence
Among such people many would withstand the difficulty and adhere to the easier life style, but also many would accept you if you can bring a clear evidence to them. One example for this can be that a verse of Quran would affect their hearth so that they would become curious to know more about it, what is it, what is it talking about. One another way to make them curious about the religion, a monster (in their new perspective) that they have once escaped from it, is to ruin its obstacles, such big religious claims that they have once used as excuses to call the religion that monster, misleading, nonscientific, non-intellectual, and etc. To do so, we can try to show them that such claims are not necessarily unreasonable, this may bring them back to the debate desk, it is only then that we can go through the first more fundamental way addressed in the first part of this answer. Two reasonable person can debate based on reasons and this needs them not to be pre-guarded hardliners. TO make them put aside their guards, we can ask them first some questions, since people don't like to be answered by you (against your teachings they have a guard) but rather like to reach a conclusion on their own, this way they will feel safer. I will try to list a few questions that may help disarm your friend in such a discussion (of course it would be funny for him to be expected to directly become a believer in Imam Mahdi, based on our explanations, while he still deny Allah, so that here by no means we want to prove Imam Mahdi but only to reject their claims that it is basically impossible, at least we want them not to think of their own claims as obvious and absolute):
if someone is living people barely ask him why he is alive, but rather if someone passes a way it is very common to ask what was its cause, so that a simple question would be this: "which one requires a cause, remaining alive or dying?" Of course, this is not a very detailed question, one may answer you both needs their own causes, for example aging is one common cause of death and remaining alive for a long time needs somehow to overcome this barrier, that is, remaining alive also needs a help, a cause. If you are answered by such an answer, then you should ask him what age is acceptable to him as average before death, he may answer a number between 60 to 85. Then you can ask him if he would then wonder and deny if he hear in news that a man is 120 years old? He would say No, that was average and this is a rare event acceptable in the same statistical system of data. Then you can ask him what about 150 years or 180 years or beyond, of course he cannot easily reject the possibility but would only say there are lesser possible compared to the 120 years in the former example. Then you would ask him what would be the probability of a person to live about 1000 years, he would say almost impossible, but that word "almost" in his answer would be a great gift to you! This very small (but not zero) probability is according to the system of data available to us from scientists and etc., such a small probability states a very very ... very small (the number of very's here depends on how much small that the probability is) number of people throughout the history may live over 1000 years, and religion also is not claiming that the number has been a big one, and particularly here we are talking about one of them, Imam Mahdi --peace be upon him--.
The second question would be inspired from the answers you are given from the previous question, there your friend would say ok it is possible but doesn't seem to be practical in the real world, since science has never recorded any such a long life before. You can answer him the new science has not recorded them but history has recorded some, Noah --peace be upon him-- being one accepted by all the heavenly religions. If the possibility is too small no one should expect such an event to happen every two or three years or even decades or even centuries for the scientists of the new era to record them easily, you don't agree? Also one can easily ask this way: "if Imam Mahdi AS was over a thousand of year old and once you understood it will it then be the first thing that you thought about as impossible but then you have found it just true?" This question is addressing the position of the new science, there are many witnesses in history of science that something was considered as impossible but then it was experienced and proved to be very possible and even very common in nature! (Breaking the sound wall was also considered as impossible based this science!)
now a deeper challenge is to discuss about the biological and psychological aging processes and senescence, and also about the scientific meaning of the Time, about the hidden dimensions of this world and things like that, some being considered as scientific issues and some only as semi-scientific issues (although it is still a place for the scientists to play in)!
About the aging processes and senescence: Isn't it possible for the rules governing the aging processes and their mechanisms to be discovered scientifically? Don't you think one day science may be able to discover them and accordingly prevent them through some medications? One may even try to find a gene which controls senescence and alters it to obtain a state of eternality! Now let me say this has some witnesses in Islam, a specific fountain is reported to exist drinking whose water can modify the human body's fine substructures such that the body will become diseases-proof and senescence-proof, say somewhat eternal. (Like about Khidr --peace be upon him--, also "Saad" (ص) in [38:1] is stated in Hadith to be the name of such a fountain under the right pillar of the Throne.) Or in a Hadith [you can find it in the book "Elal-ul-Sharayi'" written by Shiekh Saduq] it is stated that once Jesus --peace be upon him-- advised a man's wife not to eat to the extent that her stomach is full and this will slow her face skin's aging process, the couples were going to divorce due to this problem. That is, eating is one important key to the aging process, according to the religious scripts, then we have the issue of heavenly provisions, examples for it in Quran is: I. for the Virgin Mary "... Every time Zechariah entered upon her in the prayer chamber, he found with her provision. He said, O Mary, from where is this [coming] to you? She said: It is from Allah. Indeed, Allah provides for whom He wills without account." [3:37]; II. for the disciples of Jesus "Said Jesus, the son of Mary, O Allah, our Lord, send down to us a table [spread with food] from the heaven to be for us a festival for the first of us and the last of us and a sign from You. And provide for us, and You are the best of providers." [5:114] ; and III. For the Israelite "and We shaded you with clouds and sent down to you manna and quails, [saying], Eat from the good things with which We have provided you. And they wronged Us not - but they were [only] wronging themselves." [2:57] (also cited in [7:160] and [20:80]). And there are witnesses that Imam Mahdi --peace be upon him-- is frequently fed with such heavenly provisions, so that maybe someone would claim that such a diet is the cause for his extraordinary long life. Some also may claim that he, with whom is the knowledge heritage of the holy prophet, knows how to prevent the aging processes and this is more acceptable to a Shia Muslim to be more precise. Of course none of these claims have scientific proofs today, but at the same time none can be regarded as impossible, thus, are worth mentioning for someone who cares to know the truth, since none of the possible answers should be removed from the desk.
About the conception of the Time: do you know what exactly the definition of time is? Do you know any scientist who can claim to know what is the Time intrinsically? How is the perception of time and its passage, is it logical or it depends on the person who is to realize and measure an event and on the conditions for that measuring? That the relativity theory claims (
every rigorous proof based on a claim is still nothing more than the claim it grew upon) a relation between speed, mass and time does show that the modern science is trying to understand the Time but yet has nothing to say about the Time based on a firm ground! There are many debates to define an arrow of time, many debates to understand the reversible and irreversible phenomena, to understand if time is really another dimension of our universe, and many other unanswered questions. Islam has also an idea about time, I don't claim to have fully understood this idea, but it seems that one can intentionally change his perception of the Time. The Relativity Theory may be able to explain this using one of the following ways: FIRST, the more faster to travel the slower passage of time would be sensed and measured (the ascension of the holy prophet to the seven skies, in his material body, is known to have happened during only a few seconds, travelling the whole universe in the time it takes a bowl to drop, during this travel the prophet was seeing what he was seeing through his hearth: The heart did not lie [about] what it saw. [53:11]); SECOND, according to the general relativity theory great masses (or equivalently energies) can shape the space-time manifold, causing a curvature in the spcaetime that in Shia school of thought's language is called folding up of the earth or the skies [at least for the case of Tay al-Arz and Tay as-Samaa'], then moving in the background uncurved space would be like moving a longer path in the wrinkled space, and if it is true also about time (and not only space) this can scientifically explain changing of the perception of time and aging without even requiring a motion at the speeds comparable to the speed of time. This latter explanation is more acceptable to Shia Muslims. You may ask about how to prepare this much mass or energy, that would be the science that our present knowledge has nothing to say about but of course no one can claim it to be impossible, or logically wrong. There is a Hadith that states Imam Mahdi --peace be upon him-- would live no longer than Abraham --peace be upon him--, that is, about 180 years, but his time is different than ours, the Muslims. A witness for this time being different for different consciousnesses is this verse in Quran: "And they urge you to hasten the punishment. But Allah will never fail in His promise. And indeed, a day with your Lord is like a thousand years of those which you count." [22:47].