I'm tight/stiff in my Achilles, no pain yet. Will a week off running be enough to stop it from becoming an injury? Is the stairmaster Ok instead?

Possibly. It is possible that if you have an overuse syndrome that involved your Achilles and is related to your running, that rest would help. A stair master would actually likely be harder on your Achilles that running, so I would avoid that. A bicycle would probably be a good alternative to give your Achilles some rest while you maintain your fitness.