Yes, they eat. And not just mice. Rats are wonderful hunters. They swim well, can dive and fish in fish, shellfish or amphibians. Also, these rodents move fairly quickly, so if they wish they can catch up with a small animal.
Often the victims are small rodents. And in urban conditions, only mice live. In addition, especially dexterous and hungry individuals catch even birds.
In general, these are very dangerous predators, which are not so important, what or whom there is. It is important to find food and use it to live on.
It should be noted right away that mice are very afraid of rats and try to avoid contact with them and, if possible, not even approach them. And it’s no accident. Although there is no open enmity between these two species (as, for example, between dogs and cats or cats and mice), but nevertheless it is impossible to name such rodents as partners or neighbors.
But rats will not attack mice just like that. They, in general, practically do not do anything just like that, because they are too smart.