Ok, it's a 1970 hub. Probably just some light oil in the port will get it right, if it isn't already. Looking at your shift cable, it used one or two pulleys attached to the frame. Where the the cable is sleeved, it ran from the shifter on the handlebars down to the frame, where it had a stop bracket. Then the cable ran back through a round wheel, or two, to the stop where it hooked up to the chain pull. It probably had a little turnbuckle or barrel nut to adjust the tension.
It'll probably be easier to get a new shift cable that doesn't need the pulleys, because the jacketed portion runs all the way to the stop bracket at the chain pull. Then just a couple clips hold the cable to the frame, or no clips if you shorten it up with a shifter on the seat stay. The new cable may come with what you need, and/or you can make up the connection at the chain pull somehow.