“but, except, besides”?

I did nothing that day besides eat and sleep.

It would not be as common as the other two.

It's worth noting also, the question of whether one can use beside here instead of besides. Besides and beside were once pretty much interchangeable, but have moved apart (along with a few uses dying out entirely). Unfortunately, people don't quite agree on just what way they moved apart, and as such some would say beside was just as good and of those a subset would use it naturally, while some others would argue that this is wrong (or at least obsolete), and can only be besides. It may be well to favour besides in your own use, but not be critical should someone choose beside in the same place.

Both the bare infinitive and the gerund can be used with all three of the prepositions you mention. Your example, for instance, could also be expressed as "he cares about nothing except going out".