Avi Feygin
アビ・フェイゲン
Would be the closest approximation, if it's pronounced the way I've heard it.
I think what Mike Cash meant is that the Y in your name is not pronounced like the japanese Y, since it is only used as part of a vowel sound. Your first name could probably be written a number of ways, depending on how you would prefer it pronounced. アビ is the simplest way to do it.