Good idea to open the A/C right up and besides checking connections, give it a good cleaning. Can you read any nameplate data for it. Does the fan itself work? Could be there is low refrigerant due to a leak somewhere.
I think your gen should run it fine but any motor at startup draws three times the amount of current that the name plate says. For example if the nameplate says it takes 12 full load amps to run then you are looking at 36+ amps till it comes up to speed. Could be a seized compressor if it won't start without tripping.
What is the make and approx age of the unit?