Can I use my air compressor to clean cat hair from the air coils on my air conditioning unit ?

I would not suggest doing so... there is a chance if to stronge you could damage the coils.

Vacuum cleaner dusting brush is the safest bet.

If you have a great deal built up, check your filter to be sure it is in one piece, the right size, etc. I've seen folks who have the wrong sized filter end up with tons of pet hair on their indoor coil.