Nice classic bike!
It's possible, but unlikely, that the problem is at the shifter.
More likely, you need a new cable and housing and/or the rear derailleur needs servicing.
However, you can start by
watching this Park Tool video to get a better idea of what's involved in getting your shifting to work.
It might help us if you could post a few closeups of the rear derailleur so we can see the condition.
The first step is to create lots of slack in the cable to the rear derailleur, either by slacking it off at the cable adjusters or by disconnecting the cable. Then, you can try moving the derailleur in and out with your hand to make sure a limit screw isn't stopping it from reaching all gears. You also need to check that the derailleur hanger is straight. When you've verified that the derailleur moves freely in both directions, you can proceed with the next steps, as per the Park Tool video.
With a bike that old and neglected, I personally would want to go through every element of the drivetrain to ensure it's within specification and operating properly. So, I would remove the cables and housings, service the front and rear mechanisms, clean up the cable adjusters, lubricate everything, etc. If you're new to bike servicing/restoration, however, it's probably better not to pull everything apart, as you might end up with too big a puzzle to tackle.