Is the dome light supposed to work for the passenger door on an 81?

 

Yes, either the plunger is sticking of the connector at the back of the switch is corroded. Shoot a little PB Blaster on the plunger and work it in/out. If that fails, you need to remove the rocker panel cover and pull the carper back to access the switch. In case your referring to the passenger map light, check the bulb.