I think the cat will be fine. If he's been stressed by the last 24 hours (certainly sounds like it), he probably wants a safe space where the resident cat and dog can't get to him. If you have a room to spare, put him in there with food, water and his litter box and close the door. He'll eat when he's ready and cats pretty much understand what a little box is for. You should only worry if he refuses to touch a single scrap of food for more than 48 hours, in which case, take him to the vet. Otherwise, he'll be just fine.

Its so great that you took him in, enjoy your new cat! =D