Even if you fix the PSU and that's all that is the issue and the motherboard itself isn't fried, something that old could have other stuff die at any moment. Given the randomly rebooting, it's probably a miracle it lasted as long as it did. Sounds like the data is what you really care about.

I'd strongly recommend doing the second way - get the data off the hard drive onto an external device or up into the cloud (think there are tons of places that will do this for you if you aren't comfortable doing it yourself), buy yourself something new and copy the data back. I haven't bought a computer in the UK in 30+ years but looks like entry level towers at Curry's start between 300-400 quid, if it's not just the PSU needs replacing you'll start to approach that fixing the old one in a hurry.