There are two kinds of “sound control” problems: 1) airborne sounds, and 2) impact sounds.
Airborne sounds (talking, tv, etc.) is the easiest to control.
Impact sound (closing doors, heels clicking on floor from walking, etc.) is much harder to control. Impact sound travels through building materials (floor joists, wallboard, etc.) much faster and much more efficiently.
In order to control impact sound, you need to isolate the surrounding material. Some of these problems are “built in” and cannot be resolved easily (or cheaply).
You indicate that most of the noise comes through the ceiling of the closet. You didn’t indicate what type of noise, (talking, TV, clicking heels walking across the floor, etc.). So, I’ll give you a few items that could help both types of problems.
First, you need to understand that all materials have a STC (sound transmission coefficient) rating. Some materials are better for air borne sound and some are better for impact sounds.
For impact sounds we try to 1) stop floor joists and underlayment from extending under a party wall, 2) provide separate party walls between rooms, and 3) stop roof joists from crossing from room to room. Obviously it’s too late for those techniques, so now you need to consider techniques for each individual problem.
For upstairs doors, I’d try: 1) provide rubber bumpers on jambs, 2) provide weatherstripping to cushion the door when closing, and 3) add closer to control speed of door closing. For clicking heels on floor, try carpet with extra thick carpet pad, etc.
If you don’t have access to the upstairs, add sound board ( fiber board) that @Hndygrl suggested on the ceiling.
I’d also seal your closet doors with weatherstripping to keep the sound in the closet.
If you have a light fixture on the ceiling, you could consider moving it to the wall and sealing up the hole in the ceiling.
Again, all materials have a STC rating and I’m sure you could Google them and their STC rating.