How big an air compressor

Watts is not what you need to be looking at. Small generators are rated in volt-amps( VA), and with a motor the VA draw will be higher than the watts. The difference varies, as an approximation I'd add 10% to the watt draw of the compressor to get the VA. The linked-to HF compressor specs say 2.6 amps, so at 120 volts that would be 312 watts, increased by 10% would give 343 VA. I'm skeptical of the specs. 312 watts converts to .42 horsepower, so the motor in that compressor would have to be 79% efficient for the specs to be correct. I have a compressor with 3x the air delivery specs of the linked HF compressor and when I fill a tire it spends a lot more time pumping the tank up than delivering air. The HF compressor might be good for emergency use, but for general use I'd consider it pretty lame.

I'll add that the eu2000i will run a different 1 HP compressor I have, but just barely. And the generator sometimes overloads when the pressure switch kicks in at 80 PSI, so I have to prevent it from restarting until I draw the tank down further.