"Faculty" is a venerable term for a university division1, still used by some schools either in place of or alongside more modern designations like "department" and "school"2.
In all cases (Faculty of X, X Department, etc.) you could say either
I'm a student in [Your Division] at [Your School].
or, if your division has its own "name", such as a named College or professional school within a larger university,
I'm a student at [Your Named Division] at [Your School].
You could also say
I'm a(n) [your subject] student at [Your School].
Your specific example doesn't work, because as far as I can tell, the University of Oxford (the famous old one in England) doesn't currently have a Faculty of Arts.3 However, one could say:
I'm a student in the Faculty of Music at the University of Oxford.
I'm a student in the History of Art Department at the University of Oxford.
I'm a student at the Ruskin School of Art at the University of Oxford.
I'm an art history/art/music student at Oxford.
The first set are quite formal, suitable for writing in a cover letter to a resume; the last is probably more natural when speaking, and I think would be most universally understood.
Note that you should avoid saying "I'm an Arts Faculty student", at least in the US, because "faculty" can mean "professor" so it sounds contradictory.
1 According to Wikipedia,
A Faculty of Arts is a university division specializing in teaching in
areas traditionally classified as "arts" for academic purposes,
generally including creative arts, writing, philosophy, and
humanities. It was one of the four traditional divisions of the
teaching bodies of medieval universities, the others being Law,
Medicine and Theology.