Is it a good idea to send an e-mail to my future manager and request more details about the project?
It's a great idea!
You can demonstrate your continued interest, signal your desire to hit the ground running once you start, and be even more prepared for a fast beginning to your career. What manager wouldn't want to see that sort of enthusiasm?
You really shouldn't worry about what sort of expectations of your performance you are setting - you want you new manager to have high expectations, and you want to be in a position to deliver on those expectations.
If you have already signed a Non-Disclosure, then it should at least be possible. But your company may still not want to give you more details until you have been through the complete onboarding/training process, and are actually on the payroll. It's also possible that the project details are still in flight, so sending them to you may not be as useful as hoped.
Still, it's worth asking. If you sense any hesitation on your manager's part, you can always back off of your request. And you'll still get "brownie points" for showing how eager you are.
I once made a similar request after agreeing to join a new company. I got some interesting work to do in my spare time leading up to my first day, and even got to design and lay out a new computer room. The week I joined, I felt like I really belonged.