Thank you for the kind replies!
It doesn't have a bottom because I wasn't sure how I'd wiggle it onto the cup over the handle (one of the ends is a bit too tight), but that's a great idea.
I'm wondering if you made the bottom as a circle, and then the cup part as a rectangle (then sewn to the bottom) that doesn't connect to itself, but has some kind of tie so that the two ends of it can meet under the handle? Does this make sense? haha it's a little tricky. But if this method worked, then it could also be less tall, as it wouldn't have to go all the way up to the top of the handle, and you'd be able to drink out of the cup normally without eating yarn fuzz.