There are very mixed reviews about cats and milk. Cats usually will drink milk, but it may not be the best substance for their digestive systems. Many cats are actually lactose intolerant. So giving milk to your cat may result in him/her having an upset stomach or diarrhea.
An alternative is a product called Cat Sip. It's a milk product made for cats, so your cat should be able to digest it easily. You can pick up Cat-Sip (or other products like it) at your supermarket in the cat-treat section. It looks like a little juice box.