Actually, a car's A/C compressor turns significantly faster than the engine, since its pulley is usually smaller than the crankshaft pulley. 540 RPM would be equivalent to a crankshaft speed of about 300-350. Of course, you could employ the same trick; use pair of differently-sized pulleys to raise the RPM.
And most modern A/C compressors don't cycle on and off, but operate at "variable displacement", meaning a valve opens and closes variable to move a fraction of what the compressor is actually capable of pumping. But it's not a small fraction, so even if you could get the compressor to turn at, say, 1000 RPM, I don't think you're gonna get a lot of output.
There are multiple other challenges. Where do you put the dryer? Or whatever they call the part that sits in front of the radiator of the car? It's usually pretty big, about the same area as the radiator. Air has to pass through it pretty quickly; that's why the car's cooling fan is programmed to turn on whenever the A/C is running.
The BIG challenge, though, is that the A/C is usually integrated in to the car's powertrain control module (the engine computer). "Weaning" it from the PCM, to run on some mechanical control, would be a project.
One thing I've seen -- I am NOT making this up -- some folks with old beater cars have come up with "ghetto" A/C units using -- don't laugh -- Igloo-type coolers full of ice or those gel ice packs, and fans. You cut an opening in the lid of the cooler and mount one or two of those 12VDC fans like you see in desktop computers (they make bigger ones, too), and wire it to the car's electrical system. I know this sounds totally "you might be a redneck if....", but it's been done, and apparently it's pretty effective.