Thanks for the recommendation :). I have solved it and I was able to update from 1.71 to 5.1 no problem. The main reason to upgrade was that the new server set up (php 7.2) didn't let anyone log into the admin panel.
Here is how I did it:
- Backed up the site
- Updated first to 1.7.2 by replacing /include and tried to log on (no luck)
- Updated to 2.0 by replacing /include and copying over /addons and /themes, then tried to log on (no luck again)
- Updated the same way to 3.0 and then 5.1, admin panel started working.
The old theme is a little broken, but I think I can fix it. Everything else works fine.