Generator can only run AC with everything else off

What size (btu) A/C?

If it's 13.5K btu I would think that a generator with a 4500 watt surge rating would handle the A/C start-up while running a few other low power items. On the other hand starting the A/C while the microwave is running would probably be enough to activate overload protection on the generator. If it's a larger A/C unit (15K btu or higher) then it wouldn't surprise me that everything would need to be shutdown to get the A/C started.