Well, if you’re using the .deb package for Ubuntu, I have no idea. But on Fedora Linux all you have to do is download the latest .tar.bz2 file and unarchive it right over your existing install (on my machine /opt/sublime_text_3). Then start up sublime and you should be good to go.
Thanks to the publisher for not documenting this anywhere. There’s nothing I like more right at a project deadline than going on an easter egg hunt and finally just using trial-and-error to come up with an answer.