Hello, I think you may be getting a bit ahead of yourself on the diagnosis here. First let's start with the problem.
I assume based on your description that the a/c is not getting cold and the fan does not come on 1st speed when "AUTO" is selected. Is that correct?
The first and most likely thing to check here would be the refrigerant charge. It will require an a/c machine or gauges to check. When you hook up to a system you should have approximately the same pressure on the high and low side and it should be a little higher than the ambient temperature. So on an 85°f day, you should have approximately 85 PSI pressure. With the vehicle running and a/c active the high side should cycle up to about 175 PSI and the low should cycle down to about 35 PSI.
Typically a low charge is the most common reason for the a/c not to get cold. The pressures when you hook up are a little lower, when you evacuate the system it will be about 1/2 lb low on charge so the low pressure switch will not allow the a/c clutch to engage. Once evacuated, held in vacuum for a period of time (dries the system), and recharged with the proper amount of refrigerant the system works well again. I advise you DO NOT add a can of refrigerant from an auto parts store as it does not do a good job.
If you can't get it to a garage to check the a/c, but you have access to a VW/Audi scanner (VAGCOM) you can check for stored DTCs (Diagnostic Trouble Codes) in #8 climate control
. There should be an associated code to go along with the inoperative a/c.
If the problem is not what I have described, and there is something else going on, please let me know along with any other details you may have or things you have done and why. The quality of my answer is directly related to your description, so please be a complete and detailed as possible.
Jake "The Audi Doctor"