I'm trying to remember how we stepped down from pred and onto Rimadyl (also an NSAID) but it was a long time ago and my old memory cells are misfiring. He's only been on the pred for the 10 days? The weaning schedule seems reasonable for such a short duration of treatment, but I have no idea how long the wash out time for pred is or at what levels (if any) the two drugs would interact to cause an adverse reaction in a dog. Unfortunately, you may have to wait till tomorrow to get your answer when you talk to the vet.
13 years!! That's a wonderful age for a labby! Our Luke was 15 when he died and still pretty chipper right to the end. Labs are such cheerful souls.
He had arthritis, as well, but we were able to manage it with glucosamine/chondroitin supplements only.