What rim is it?
If it's a single wall rim, it very well could be the rim and not a trueness issue. Double walls are harder to fuck up THAT bad, but can still happen to them.
Look at the photo below
Let's say your wheel is hopping up at the circled parts, you would want to tighten the spokes JUST in the circled parts, and loosen them everywhere else. So tighten the circled spokes, but loosen the rest. Starting at the valve makes it easier, but locating where it's hopping is tricky without a truing stand. At home I just eye it up in my frame as best I can, or use zip ties and get it as best I can. Up and down is a pain in the ass but taking your time and doing 1/4 turn on each and being patient will go a long way.
It's best if all your spokes aren't piss tight either to start with, if they're pretty damn tight, loosen them a tad so you don't risk messing the nipples up or snapping a spoke. Also make sure your rim is fairly true side to side, you don't want it completely straight because when you go to do the up and down, you'll throw it out of true again anyway. So get it fairly straight, then do the up & down hop, then fine true it side to side.
Also keep in mind you said it is an old/beat rim, so it will never be 100% true, so don't be surprised if you can't get it 100% after spending a lot of time on it.
