I've been using GoPros since version 1, although my most current one is a 10.
The first thing to consider is what mount you are going to use. GoPros aren't designed very well for handholding. A nice thing about the GoPro 12 is that it has a 1/4" threaded hole at the bottom, so you can screw it onto a selfie stick if that is what you want. But I don't think Tokyo
Disneyland allows selfie sticks. Maybe a short handle mount would work. Or a wrist mount. Or you could handhold it by setting it on the edge of your teacup and just holding it there.
For the carts, you'll want something like a helmet mount or a chest mount or maybe even a handlebar mount. You might want to contact them in advance to see what they recommend. I've also found that a lot of places like that have mounts already built into some helmets or on the cars or whatever.
As for camera settings, that's a bit of a tougher question to answer. The biggest problem you'll face is that GoPros are small-sensor action cameras. They are amazing with a lot of light, but not as amazing in low light. The Beauty and the Beast ride is pretty dark, so you are going to struggle to get good image quality regardless of the settings. You might want to watch the video below.
One thing I disagree with from the video is his suggestion to use hypersmooth if you don't have a gimbal and to raise your shutter speed to 1/192. I think you'd be better off with hypersmooth turned off and your shutter speed set to 1/48 (at 24fps). You'll get 4 times as much light at 1/48 and I think that is more important than hypersmooth. It will be shakier, but the image will look better.
Have a great time in Tokyo! And enjoy being able to ride the Song of the South version of Splash Mountain. Boy are you in luck! If you have any questions after watching that video, let me know.