It doesn't matter how old the phone, computer or iTunes installation is. Only the date of the backup matters. Just make sure to manually take a fresh backup of the old phone to iTunes before you setup the new one. (You should be running the latest versions of iOS and iTunes, too).
I'm on my third iPhone and about to get my fourth. The setup and restore from the old backup has always worked without a hitch. That's one of the big reasons I've stayed with iOS devices despite the lure of Nexus and Galaxy. I hate the idea of the walled garden but its seamless transition to new devices is a narcotic I can't seem to quit.