Yes, you can treat a dog with both Prednisone and aspirin. But some Veterinarians do like to take a hip X-ray prior to administering Prednisone for arthritis and similar situations. Prior to giving aspirin they may want to a blood test to insure the health of ones animal. (bleeding disorders)
It is best to see a veterinarian for both medicines due to the side effects. Prednisone is a strong medicine for allergies and Benadryl is generally tolerated better for this usuage and can provide relief.