How long can ATC operate for without main power?
Nearly all generators designed to serve as a backup power source in a safety critical environment (Hospitals, Air Traffic etc) will be designed to run 24/7 "indefinitely" with hot refuelling. 

Obviously there will be a real world practical limit to that, but it will be many days if not even weeks potentially depending on the generator.

I suspect there will be huge variation in the type of standby equipment available, pre-loaded fuel and so on depending on the mission. For example, it would be entirely possible to have two generators which can switch over - allowing the other to be serviced and maintained.*

*NB: I don't know if this is an actual "thing", but I'd expect it in certain circumstances, for example, military.