My puppy fell down stairs and is limping. What should I do right away?

Poor Hamish! I'm encouraged to hear that he is bearing some weight on the injured leg, as this makes a fracture seem a bit less likely. Instead, it sounds as though he may have a soft tissue (muscle, ligament, tendon) injury. Keep Hamish restricted to a crate or very small room to prevent him from running, jumping or playing over the next few days. Try to limit his activity to short leash walks strictly to eliminate. If he is still favoring the injured leg in the morning, Hamish should be examined by his vet for further evaluation and possible anti-inflammatory pain medications. Do not give any human pain medications, as none are safe for dogs. I hope that he feels better soon!