My through-the-wall room air conditioner (2 years old) turns on and blows air, but the compressor doesn’t kick in for a few minutes and then, when it finally does, it will only run for a few minutes before shutting off and going back to just blowing ambient air. Any help appreciated.
Could be a number of things. As far as thing that could be causing this that you could repair yourself. For starters I would check and make sure the outdoor coil is clean and the outdoor section is moving air as well. What size unit do you have?
Possible bad capacitor. 
Compressor cutting out on the overload-could be pulling to many amps need a/c technician