I have a switched ceiling light that was originally intended to be a ceiling fan. It has a black a white and a red (which is capped off) wire at the light.

There are 2 switches in the wall. One operates the light and the other simply has a black and a red hooked to it and does nothing.

My question is... I would like to install another light in another part of the room, controlled by the switch that isn't switching the original light. Can I take power from the ceiling box using the red which is capped off and another of the wires (black or white?) or do I have to run another wire to the switch somehow?

Ideally I would like to take power from the ceiling box, and run it to the new light and have it switched by the switch which currently isn't switching anything. Is this possible?

Yes what you would like to do is very possible.