Absolutely not! Most OTC pain killers for humans are very toxic to dogs, and really the only one on the safe side, is aspirin. This too can be dangerous. My small dog was on baby aspirin a few years ago (with her vet's blessing), and took one in the morning and one at night and only with food. In less than 2 weeks she developed a bleeding ulcer and nearly died.
Being that the puppy is so small at 5 pounds, I would not give it anything without first consulting with a vet to see what is safe and also what dosage - if any. By playing doctor, you could do much more harm than good.