Just like it says in the title, if I customize the kernel for the computer, does it stay that way if I were to update to 12.10? Thanks!
The largest condition is the version you're running. If your kernel still newer than the latest version in Ubuntu — eg you installed 3.6 before upgrading to Precise, and it's packaged in a similar way to Ubuntu's kernels — you'll stay on your custom version.
If you installed a custom version of 3.0, or whatever was running on 12.04, and you hadn't done anything special with the versioning, grub will likely rank the newer kernels higher. That's just how it works.
Whatever happens, the upgrade shouldn't remove your custom kernel. It should still be available in grub and you can remove the newer ones after (if you want to keep using your older one).
Note: it's not recommended to stick with an old version of the kernel without applying security updates. If you're going to run an ancient version, make sure you're applying patches and recompiling the kernel on a regular basis to remain safe.