Hi, P.. Sweetie, don't worry that your child has no appetite and is not active during the time that he's sick. That's natural for children or for anyone else. Think about how you feel when you're sick -- you probably don't have the same kind of appetite that you do when you're feeling fine. He probably has a sore throat and can't swallow anything comfortably besides milk. Even that may hurt him if his little glands are swollen. And he's not going to have any energy for playing when he's sick either, just as you do not have any energy when you have a cold.
His milk is fine for now. It's got protein and vitamins, so he will be OK. Try giving him some water as well, maybe some juice that's not too acidic. Maybe grape juice or apple juice. That way, he can get more liquid and less mucus.
I would advise you, though, that if the cold lasts longer than 10 days or so, to please take him to the pediatrician because in winter, colds can develop into something worse.