You have to remove the glovebox and the bit of carpet above where your feet would be and the connector is up there somewhere. I didn't actually do it myself but was there when the mechanic did it.
By a process of elimination (using wiring diagrams) he found the wire in the engine bay which controls the compressor engagement and followed it back until he found the bad connection. It's all been fine since then (touch wood).
I would check all the simple things first though before you go taking things apart (i.e. refridgerant, relays, fuses etc.). I had exactly the same thing when I initially took mine to Halfords - the fella put his gauge on and it said 'Danger - Do not add refridgerant'. This was because the compressor wasn't working so the reading was high. Actually i had the right amount in there.
Kwik fit will put the car on a more sophisticated machine but at the end of the day they are not air-con experts so if it's anything more than a simple re-gas they won't be able to help you (although they probably won't charge you either and at least you'll know if your system has the right amount of gas).
As for you pics I must admit I haven't seen a compressor like that (mine is the one with the protruding metal clutch). I have heard that some manufacturers use compressors which run all the time and use other parts of the system to cut in and out but I didn't know ALFA used them.
I think the ALFA compressors are made by SANDEN (see here for list of parts and pics of different compressors they make for ALFA's http://www.eclima.com/ITALIAN/ALFA%20ROMEO.pdf
) and it could just be that the clutch mechanism is recessed on yours so it's not so obvious.
Anyway if it's working you should hear a pretty loud click when you press the A/C button and the revs will dip a bit. If you don't hear it engage then it could be a gas or electrical problem.