Who was the first person to lead prayer in Madina? Was it Prophet Muhammad(Peace be upon him), or was it any Sahaba? If so, what was their name?
Note that the mosque in medina was only built after the hijra of the Prophet (), so one can only tell (assume) that the sahaba or sahabi who were/was sent to teach the people of Yatrib about Islam and the Qur'an were/was the person who used to lead the prayers.
In sahih al-Bukhari we may read:
The first people who came to us (in Medina) were Mus'ab bin `Umayr and Ibn Um Maktum who were teaching Qur'an to the people. Then their came Bilal, Sa'd and 'Ammar bin Yasir. After that 'Umar bin Al-Khattab came along with twenty other companions of the Prophet. Later on the Prophet (ﷺ) himself (to Medina) and I had never seen the people of Medina so joyful as they were on the arrival of Allah's Apostle, for even the slave girls were saying, "Allah's Messenger (ﷺ) has arrived!" And before his arrival I had read the Sura starting with:-- "Glorify the Name of your Lord, the Most High" (87.1) together with other Suras of Al-Mufassal.
The hadith was interpreted by some that Mus'ab ibn 'Umayr came first then Ibn Um Maktum and so on. Ibn Ishaaq also quoted that Mus'ab ibn 'Umayr was the first who was sent to teach the new Muslims in Madina he went in the company of those twelve representatives of al-Ansar (see for example here in Wikipedia).
I found in Shi'a sources that Musa'b ibn 'Umayr was the first to establish jumu'ah prayer in Medina following the order of our Prophet () (see here in Arabic) this can also be found in ahadith which are considered as weak compiled by at-Tabarani (see for example here in al-Kabir) and ad-Daraqotni another narration indicates that the first person who lead jumu'ah was As'ad ibn Zurah (see for example in sunan abi Dawod) basically this might go along with the other view, as Asa'ad ibn Zurarah assited Mus'ab in his preaching.