The same thing that the cloud company would do. If there are other people there who share that guy's responsibilities, have them do it. If there aren't, you should have an on-call.

Cloud just outsources that problem to another business. Sure, they have better reasons to actually cover those positions and make sure they have on-calls and backup and a disaster plan, but just because you pay extra money for it doesn't actually make it work better if the company underlying it sucks.