Die of asphyxiation
Onto a more serious... Answer... Well, the previous thing was actually the most serious...
Now, onto a more fun answer!
I'd sleep. Alot. As for the times I wouldn't be sleeping, I'd work on a Moon-Rock Palace! Or make scupltures out of moon-rocks. I'm sure I'd get good at it at some point, if I'm really stuck on a moon for a thousand years without getting hungry, thirsty, or needing to breathe.
I could also work on some gigantic "painting" on the side of the moon, to confuse the inhabitatns of the planet the moon is circulating around, if there are any! I say "Painting", 'cause knowing me, I'd make something akin to what a hungover dog that's heavy on painkillers would make.
Actually, ah'd probably just do what I've done throughout my post for the whole thousand-year period. Ramble. Ramble my mind to shattered pieces, given that I can't make sounds in the "vacuum" of space.