If the cones are pitted, the bearing surfaces inside the hub shell may well also be damaged. As the rim is not particularly valuable, your local bike shop may be able to sell you a complete replacement rear wheel to suit the screw-on 6-speed block you have.
If you really wanted to only replace the axle, by 1990 I would expect a non-vintage hub to be using ...
Start by checking the hardened bearing surfaces on the inside of the wheel's hub. If they're not smooth under all the grime then its generally not economic to proceed.
Otherwise, you can do this.
Tools you will need to buy or borrow:
a freehub removal tool
two cone spanners of a size to fit your new cones so probably 15mm but other sizes are possible. You ...