My S23 ultra broke but still work

In most cases whatever initial damage you see from the accident is the extent of the problem. In some cases either the display or the touch screen will get worse over time, but chances are good your screen will continue to work fine until you can get it replaced.

Yes, you definitely want to put a screen protector on it, although one of those cases that fully encloses the phone might be a little bit better as it will fully insulate the damaged screen from any kind of contact. Either way that should keep your phone in one piece until you get the screen replaced.