You could try, but I'm not sure how satisfactory the results would be. Aluminum is often anodized prior to painting, but you would still need to use a zinc chromate primer for best results with regular paint. For a really good surface, you would probably want to use an epoxy type paint, and I believe that epoxy would take its own special primer. And for best results, you would probably want to disassemble the computer and extract the metal part that you wanted to paint.
The surface is actually hard anodized aluminum. By its nature, the anodized surface is just a few ten thousandths of an inch thick and hard but brittle. For that reason, it will develop microscopic cracks under a heating and cooling cycle which will allow for the intrusion of reactive substances, in this case individual body chemistry. Aluminum is very active and therefore very reactive. The anodized surface is basically aluminum oxide, which is generally very protective. However, for a few unfortunate individuals, the sweat can penetrate the surface through the microscopic cracks and produce the characteristic black pitting.
The anodized surface is generally more durable than a painted surface would be--that's why it is used. I'm not sure that you would be happy with a paint job over time.
So it might boil down to a protection pack, a new upper case, or the pits.
Good luck!