The Catbus eats the same food any cat would eat
AFAIK there is no definitive answer from Miyazaki. But we can infer an answer based on observations of the world in which Mei and Satsuki encounter him. The fact that the world of Spirited Away into which Chihiro accidentally stumbles is similar in many ways and has significant crossover (the sootsprites) also suggests that we can draw on this movie as well.
In this world, spiritual and semi-spiritual entities are as much physical beings as ordinary humans and animals. Their bodies occupy space and time, are solid (although they can often dematerialize at will) and have mass. They also eat and drink.
The Totoros appear to be herbivores. We see them gathering acorns. We don't see the Catbus eat anything, although as Valorum points out, a Kittenbus eats a toffee in Mei and the Kittenbus, showing that they have a more or less mundane digestive system. In Spirited Away we see inhabitants of the spirit world eating a variety of foods including pork, fish and dumplings. Some of the food may be unusual (Lin's newt) but none of it is dissimilar to what humans or animals would or could eat in the normal world.
Based on this, we can say that the Catbus and his relatives almost certainly eat the same type of food that normal cats would eat. Cats are obligate carnivores so that would likely include animal prey of some sort. Whether this includes outsize rodents or not is unclear. The Totoros do not seem to be afraid of him, but this may be because Big Totoro is formidable enough in his own right to be able to deter the Catbus from considering him potential prey. And having formed a relationship with him, the sisters can probably be considered under his protection.
It's notable that it never occurs to Mei to be afraid of any of the strange entities she encounters. This partly reflects her adventurous personality, but it also likely reflects the fact that she is too young to truly understand danger. Chihiro Ogino, being older, finds the spirit world a much more frightening and dangerous place where being eaten is a real possibility.
I suspect that in a different context, encountering the Catbus alone on a dark night without the company of Big Totoro could be a very different experience!