Sometimes starts and stays running....

I have a 1980 hatch with EA71 and i've been having trouble sometimes with keeping it running. Most of the time everything is perfectly fine. But sometimes when i start it the battery light and door lights etc stay on for awhile then clear and it'll run fine. But when they don't it'll die in about 30 seconds. I recently replaced the ignition switch because i found that the only time the fuel pump would pump was when the key was turned to the start position then would stop once clicked back to the run position. Could this problem be my alternator slowly dieing on me? I'm out of ideas. :confused:

Thanks for any help.

-cody