At the moment I have Mint on my computer, and I want to change back to Ubuntu, but with 13.04 coming so soon, should I install 12.10 and upgrade, or are there any advantages in waiting and doing a fresh install of 13.04?
If you have already decided to use Ubuntu, you should feel free to install Ubuntu 12.10, because you can upgrade to 13.04 when it comes out.
Assuming you wait for the actual release date to upgrade, most likely there will be no problems upgrading. I do recommend reading the release notes before upgrading, in case there are bugs that might affect the upgrade. (Similarly, I recommend reading the release notes before performing a fresh install, if you choose to do that.)
It's possible that a fresh install will work better than an upgrade, even if nothing is noted in the release notes about it. But if you have any preference at all, I recommend going with your preference. There's no strong reason to think a fresh install will work better, and if something goes wrong during the upgrade or afterwards, you can always just wipe the disk and perform a fresh install of 13.04!
You also have the options of: