By all means, you have nothing to lose. Do the work under a very bright light... preferrably direct sun light so you can see things as well as possible. I have sucessfully repaired many a circuitboard with a "cold" solder joint.
If you could see how they are made moving across a "Wave Soldering" machine [tank of moving molton solder] you would understand how they can go bad after all the years of being vibrated and shook in that old automobile.

Don't get carried away and add too much new solder... just enought to get the heat to flow from the iron's tip onto and resolder the wire at each connection... and don't apply the heat for too long either. The parts on that vintage circuitboard are what is called "discreat" parts and are much larger than todays components but still try not to solder all of the wires for the same part, one right after the other. Move around from one to another component so the heat does not build up in any one because you have re-soldered all its wires one right after the other.
Good luck,
Sam