Hi Tony,
Unfortunately there is only so much that can be done with older and lower resolution footage, especially if coming from an older analogue source. However, I do have a couple things you could try:
1) When you export a sequence from Premiere, try checking the box "Use Maximum Render Quality" towards the bottom of the Export Settings window. This uses a more computer intensive algorithm to scale up footage so it will be more taxing on your computer and take longer. It won't be a night and day difference, but you may find it helps. Try it with a short test clip and compare.
2) After Effects has an effect called Detail-preserving Upscale which again is very computer intensive but tries to do a better job of preserving details and clarity when doing large amounts of scaling up. If you like how it looks you could run all your old footage through After Effects and then cut with it in Premiere. This article explains how to use it: Using the Detail-preserving Upscale effect in After Effects
3) There are a handful of 3rd party plugins that exist to solve exactly this problem. While I can't speak to any one of them in particular, I know a popular one is "Instant 4K" by RedGiant: Buy Red Giant Instant 4K | Download the free trial
Good luck!
Matt