Do I need a new tire pump or can I fix my old one?
You're not going to be able to fix something like a pump: you won't be able to buy spare parts for it and, even if you could, you could probably only install them using the manufacturer's tooling.

However, the fact that your pump and pressure gauge disagree doesn't tell you which one is wrong. Maybe your trusty old pump is actually right and your new gauge is inaccurate. Maybe your pump really is old and worn out.

If your two devices each measure in a consistent way (e.g., one always reads 10psi lower than the other), then just learn the translation between them. Assuming you're not running at close to the maximum pressure of your tyres, it doesn't really matter if you inflate to 80psi or 90 psi, as long as the results are comfortable to you. If you're happy with the state of your tyres when you pump them up to a certain pressure as measured by your pump, then keep doing that and just be aware of the different number your pressure gauge will tell you. It doesn't matter if they're not real pounds per square inch (or bar or whatever scale you prefer).

If you want to decide which one is wrong (and it might even be both!), you'll need to use multiple devices. Can you borrow a pump or gauge from somebody else? Your local bike shop, for example, probably has a pump they'll let you use.