I have an old, retro 70s shower wall (plastic with gold plating) that needs replacing, but I am on a tight budget. Can I seal and paint the existing shower wall until I am able to rip out and replace? Any advice on how to proceed?

Sure! Not saying it's the recommended deco for a shower stall but for your purpose; just fine.

Rough up the surface of the shower wall real well with sandpaper and use pool paint (if Lowes/HomeDepot does not carry it, look for a local pool supply store).