that would only be able to tell you how much total power was used, and by that, how long total it was on. you wouldnt be able to tell how many times it cycled on and off.
there are 1001 ways to do this, but personally i would use a microcontroller like an arduino board and a few parts from my junk parts pile. hook a relay with a 120v coil up to the compressor in parrallel so it triggered whenever the compressor kicked on, and then have the arduino record how many times the relay closed (or opened).
or- you can put a decent digital temp probe in there and log the data every few seconds (again, i would use an arduino and a DS18B20 temp sensor because i have them laying around). when you graph the data you will see a distinct saw-tooth pattern. count the number of 'teeth'- thats how many times it turned on/off.