if your cat had it in its mouth or clawed it, the mouse will die. There's a bacteria in their saliva and their claws that kills rodents and birds in a flash.
If you take it really fast to the vet it could survive, but its so small your vet probably won't want to touch it.
The FIRST thing to do with a baby animal is to warm it up, their intestinal system doesn't work properly if they are cold and you could kill it just by feeding it without it being warm.
Take a ziploc bag and fill it with warm water. Put a towel on top and the baby above the heat. Change the bag often! that's the cheapest way. You could put a heating lamp, or a heating pad under.
Feed it only once its warm, milk should be warm but not too much. NEVER feed cow's milk, makes most babies have diarrhea. Try goat's milk instead, or a milk replacement.
Baby mice eat very little. About 3 drops every 3h is enough if its eyes are closed.
you should call your vet or spca and ask if they have the number for a wildlife rehabilitator in your area. A lone mouse has less chances of making it than if it has other mice to stay warm against. and a rehabber has a lot of experience and will have an easier time saving the baby.