Many lines have water fountains too.
I wouldn't say
many lines have water fountains, a couple of the rides with a focus on outside queues perhaps, such as Jurassic Park, and I believe I spotted one at Dudley Do Right? Could be wrong it's been awhile since I've been on one. There might be one, maybe two more.
But anyway, there are trashcans along the way in the queue and if your drink is easily disposable they will make you throw it away before you go on the ride. I'm not 100% on if the drink is bottled, I would imagine you'd be able to keep it and store it since it's concealed, but someone else can either confirm that, or you can ask that at the park.
If you get a refillable cup you can either bring it with you and stow it away in front of you in those pouches they have usually on rides, store it between your feet / legs, or on the side (the Simpsons is the only ride that suggests you can place your belongings beside the ride vehicle). Also, on rides in which you can get access to free lockers (RRR, Hulk, FJ) is how you can store your drinks and other belongings and on RRR and Hulk you have to store all your belongings anyway. However, the smaller ones are free, so depending on how big your cup is, you might have to still dump it before placing it in the locker, however Butterbeer cups should be fine.