It passes through but it causes damage. It can knick bone, muscles, nerves and veins and arteries. To keep it simple have the surgeon examine him and say, "You're lucky, no vessels or bone. We'll clean and debride if needed." The bigger wound is on the bullet's exit (his back).
He won't be in the hospital much longer than twenty four hours and no longer than two days if he doesn't have surgery or complications. He gets a tetnus shot and antibiotics. The surgeon may choose for him to use a sling for a few days and then start passive ROM (range of motion) exercises which he will demonstrate or the nurse will before discharge. Movement will be evaluated in the follow up office visit. Here PT may be considered if patient didn't do his ROM as instructed.
Nurses will show family how to change his dressing and what sypmtoms of infection are. (Redness, stinky, pus-like discharge, or fever). Dressings can be wet-dry which involves packing the wound with the wet one and covering with a dry one. Daikins solution or saline for the wet one. Or just dry. This is simple and easy. If he doesn't have family and can't reach his back dressing, a visiting nurse can be used. He'll get PT-if he's a policeman and he's shot on the gun hand side or if his job requires manual dexterity and his recovery his slow. Muscles heal slower and hurt like heck when you move them, but you have to move them so they don't stiffen or contract.
Yes, he can shower, but he'll be told not to get the dressing wet. Hospital shower streams are pitiful and not like home. Smaller dressings can be covered with a larger plastic dressing. Otherwise, if a patient is awake and alert, can sit up and has one hand working, it's self care. The bathwater in plastic tub is brought to the beside with soap, washcloths and towels. He sits on the side of the bed and washes himself or he can stand at the sink. He's told to hit the areas he can reach and he's left to do it on his own. The nurse steps behind the privacy curtain or returns in a short time and finishes up (usually the back). I don't see why a story needs a section on a bath for an overnight patient, but now you see it's not that interesting.
Longer stays in hospital might warrant a full bed bath, but those folks aren't awake and alert like your guy will be. If he can bathe himself, he will be asked to do it.