Caroline:
Unfortunately, Disney doesn't provide any accommodations that can be rented by the hour.
However, every park has a Comfort/Baby Care Center with a dimly lit room. Although they are not set up or intended to be nap rooms, you can take your child in for a bottle and some quiet down time. (Be aware that you can't take strollers in.) Depending on the park you visit (and my guess would be you'd choose Magic Kingdom), there are always dark nooks and quiet places--even attractions--where you can go for a soothing rest stop. Here are some of my favorites at the Magic Kingdom:
- One of our family's favorite rides to catch a cat nap is the steam train. Best of all, you can ride around and around and around the park--keep in mind that its whistle might arouse some light sleepers.
- Carousel of Progress: Dark and soothing, this attraction runs about 20 minutes. One of my favorites on a hot afternoon after lunch for a cat nap.
- Hall of Presidents: Very dark. The movie runs a good 25 minutes or so. I get my best sleep here.
- It's a Small World: It provides a long ride (well over 10 minutes; 15 minutes with unloading time) with music that drones on and on....but if you have little ones that are excited by lots of colors, you may need to skip it.
You could always opt to "see" a movie at the theaters at Downtown Disney. But, quite frankly, if I were in your shoes, I'd either consider getting a room at the All-Star hotels or even an economical one right off property. You'll be able to check in late afternoon for a nap and it will allow you to spend the night and even come back the next day!
Wishing y'all a magical vacation!
Whitney