I want to install a lighting fixture in my bedroom. Can I run a wire from the outlet that is powered by the switch to the ceiling to power the light?

Yes, you can do that. You will probably have to tear into the walls & ceiling to get access to the box and to run the cable.

It might be easier to run cable from the switch box by the door up to the ceiling location. That will work only if the power cable comes into the existing switch box. If it's just a switch leg, it won't work.