I am wanting to make a small graduation cap out of gumpaste/fondant. Will the top of the hat dry hard enough so that it will not droop when I glue it on? And how far in advance should I cut it out and start letting it dry?

Yes, the gumpaste will dry rock-hard (don't use Bakel's gumpaste, though), and you can get your pieces cut as early as you like. The earlier, the better. Make 2 as insurance.

I agree, the earlier the better, it gives it time to dry hard, (especially with so much colouring in it to make it black). Also, keep it out of humidity just to be safe.

Make it at least 2 week's in advance, add a little Tylose to the paste so that it really gets hard.

If you're really worried about the cap drooping and you don't need the top of the cap to be edible you can cut a piece of cardboard and cover it with the fondant/gum paste. Make sure to keep it away from light because it will fade.