This man was Asif Ibn Barkhia the successor of prophet Solomon a.s. who was chosen by God for as successor for leadership of believers after death of prophet Solomon a.s.
This knowledge is some special names of God that any miracle can be done by using them.
In the Dua Simaat some of these names and the works that can be done by them are described.
Some of those works that can be done by knowledge of book described in this dua are:
- open the closed doors of the sky
- remove the narrowness of the doors of the heaven
- to make easy the difficulty
In short by having this knowledge every science and every part of nature and every miracle will be in your control.
this knowledge has different parts and all the prophets did not have all of it. in this verse it is said "knowledge from the book". the word "from" means he (the successor of prophet Solomon a.s. had only some of this knowledge.
In another verse Quran talks about another man who has the knowledge of book. but in this verse the term "from" is not used that means this man has all the knowledge of book. according to many different hadith this man is Imam Ali a.s.
وَيَقُولُ الَّذِينَ كَفَرُوا لَسْتَ مُرْسَلًا ۚ قُلْ كَفَىٰ بِاللَّـهِ شَهِيدًا بَيْنِي وَبَيْنَكُمْ وَمَنْ عِندَهُ عِلْمُ الْكِتَابِ ﴿٤٣﴾
And those who have disbelieved say, "You are not a messenger." Say, [O Muhammad], "Sufficient is Allah as Witness between me and you, and [the witness of] who has knowledge of the Scripture." http://tanzil.net/#13:43
Yes it can be learnt today but partially and the only required term is being pious. the complete knowledge of book is only with prophet and The Fourteen Infallibles.