To run the microwave, you'll be looking at least a 2000w generator.
If you're a/c is 13.5 or 15.5, you'll need a 3000w genny. If your A/C is a 9, you'll do fine with a 2000w
If you want to start off inexpensive, let me suggest the Costco sold Smarter Tools 2000iq (2000iq) They're $599.99 and you can always hook 2 together later on with a set of parallel cables ($60.00 approx.) and run the larger a/c units and a microwave (though generally not at the same exact time)
You run the shore cord directly to the generator. When you do this, your batteries will be getting a charge as well (nothing too great, but a little something) If you hook your generators directly to your batteries, it would take about 3 hours to bring the batteries back to 100%.
I'll add, the Smarter Tools 2000iq generator has the same Yamaha Engine that the branded Yamahas have and there is a 3 year warranty.