How do I determine which outlets are on the same circuit without power?
I'd use one of these - a tone tracer. They're mainly used by telecoms engineers to trace pairs in large multipair cables and junction boxes but they'll do just as well for this application.



Though why you don't want to pay for electricity before you move in beats me. Surely you can get a plan where you just pay for what you actually use?

I'd have to advise against doing electrical work on the cheap though, it rarely ends well.