He is a fourth-year full-time student of economics and management. It doesn’t seem natural to say that someone is “the student of the faculty.” “Form of study” is not something that sounds very natural. I understand what you mean, but that is not something that would be commonly said in English.

The first variant is good I coukd not improve on it Молодец

The first sentence is natural.

The first sentence is good, but maybe you could say "in" rather than "of". That is, he is a "student in the Faculty of Economics and Management". Or simply "student of Economics and Management". But in the US, it would be more common to refer to the Department of X rather than the Faculty of X, because there the term "faculty" more commonly refers to professors and teaching staff as a whole.