I learned then that sudo was really just a binary, so I tried to get a copy and put it in the right place. I couldn't, though, because I didn't have write permissions to it without sudo!
In hindsight I guess I could've just run the binary itself to get access, or put it elsewhere on my $PATH, or use `su` instead. Not sure if I tried those things, it was a while ago and I was pretty new to Linux. Maybe I got the file from the internet and didn't know to make it executable.
Anyway, what I ended up doing was booting up from a live Ubuntu USB and copying the sudo from the live environment to my installation on disk. It worked, and my newbie self felt like a proper hacker, fixing the unfixable. For one day I was a heart transplant surgeon :)