What is the proper way for an abbreviation of assistant professor?
In my experience, what comes before professor (e.g. assistant, associate, full) is only really used in formal contexts, and in most other cases simply "Professor"/"Prof." is enough.

And, at least in the sciences, "Doctor"/"Dr." is typically used when that is the highest title.

I've never seen both "Professor" and "Doctor" used together.

So, in your case, I suppose "Prof. X" is good enough.