A pretty long and detailed walkthrough that basically boils down to the official confirmation of what you’d expect to make sense: when presented with a blank iPhone, iTunes will offer to restore one of your backups on it (iTunes automatically backs up your iPhone when you sync it, in case you haven’t noticed). And, if it doesn’t offer, you can just force it.

As obvious as it is, I’m linking it because that’s the kind of information you like to be sure of beforehand. Oh, and you can also expect that the process will fail for a number of people. That’s how computers work.