Edward, you can remove the crank pulley with out removing the fan, radiator etc but you will need to contort your hands a lot and leave some skin on things there. There are 4 small bolts then the pulley separates in half...note the shims..that is how you adjust the belt tension. Others are probably correct that the compressor seal is probably bad and the system empty or very low on refrigerant but all you have to lose is the cost of the belt and a little of your time to give it a try.
Personally I would not pay for the labor at a shop to r&r stuff including o-rings, hoses, compressors and dryers as the parts are available and the system is pretty simple as car a/c systems go. Also, the compressor r&r is a pain in the a$$ and a shop will charge the heck out of you for it and probably break other fragile alfa stuff like the cam timing solenoid while they are in there. assuming you don't have refrigerant experience, I would do all the mechanical work then take it to a shop to vacuum down the system and charge it.
If you take it to a shop for the whole thing it will be way more than $1k...probably more like $3k.