Certainly you are not going to be able to get from the All Star Music resort to the Contemporary via the 'official' route of taking a bus to Downtown Disney, then transferring buses to get to the Contemporary. Of course I would have recommended a bus staight to the Magic Kingdom anyway, with you walking or monorailing over to the nearby Contemporary from there.
But with the park opening at 9, the buses from your resort to the Magic Kingdom won't officially start running until around 8. While they usually start before that, I don't think you will see them as early as 6:30 am.
So here is what I would do. Talk to the transportation desk at your resort as soon as you arrive. Ask them what to do. They will either find alternatives for you, or they will make sure a cab is waiting. The charge should be under $20, and you can leave your resort closer to dining time, which is nice that early in the morning!