My air compressor sits in an unheated garage. It struggles to start when it's cold. I would suggest changing oil to a 5w-30.
It was around 30 and I tried starting my generator, which is a rope start. Had a heck of a time getting it started. After it warmed up I changed oil to 5w-30. What a big difference.
You may want to read up on your generator. Your motor's lock rotor amps may be larger than your genny can safely handle
If my only way were to power up an air compressor is with a generator, I would find the smallest pulley I could get for the motor. Then I would loosen the belt a little, let it slip on start. A belt is cheaper to replace than a generator.
Can't help you much with fixing your generator.
I know I shouldn't mention this because it will definately cause a stir, but I will. I power factored my 3/4 hp well pump. It took 200 MFD's to get the job done. I had a bucket full of used capacitors, so it didn't cost me anything. If you are friends with a HVAC man, he can get you used ones from units they scrap out. It would cost you a pant load to buy new ones to do your job.
My well pump used 12.5 amps before I added the capacitors. Now it draws close to 9 amps. In MHO, my generator seems to start the pump much easier. Some people may say not to do what I did. I would definately power factor an air compressor and see if it helps the generator. Can't see how it would hurt. I power factored my pump, not to save money from the electric company which will be very little if any, but if I ever lose power, I want to use my generator to pump water.