I still own a working 1st Gen xbox360 (Xenon board, no HDMI), though I have since upgraded.
1) Keep it well ventilated. Give it at least 30 cm of space in all directions. Never put it into a cabinet, and never put it on a carpet. Hard surfaces only.
2) Try not to use it when the room you're in is over 80°F (26.6C), this will significantly reduce the cooling capacity (hotter intake, less efficient venting).
3) Don't leave it running if you can avoid it, let it take breaks and cool back down to ambient.
4) Ideally, horizontal is better (superior cool-air intake), but if you do that, again, make sure it is on a hard surface. Anything else will work as an insulator. If you want to keep it extra cool, put it on a wire baking rack, or a laptop cooling-fan rack.
Giving it room to breathe is really the only key. You can't stop the dvd drive from failing though...
That really sucks.
My sincere condolences Zell.