It's highly unlikely that you're going to be able to go to a store and match that wallpaper pattern. And, even if you could, the fact that it would be a different dye lot would make any patch obvious.

The only reasonable way I know to deal with that is to get some brown paint to match the general tone, and then add some striations with a brown marking pen. However, I wouldn't be surprised to see that area continue to peel. Good thing it's a rental unit, and not something you have to worry about over the long term.

I would see if I could smooth it out where there are little bits poking out, and color match it to a decent quality paint, and then use other color paints to faux in the pattern, so it matches, and put some curtains up on a tension rod, and pull one to each side and tie them back. The room will look nicer with them, and no one will ever notice it has been patched.