I'm also doing an upgrade at the moment in my case a Rails 3.2 project which I will upgrade to Rails 5.2, through 4.0, 4.1, 5.0, 5.1. My project does not have a lot of tests so what I'm doing now is writing a lot of tests. Mainly Integration tests because there already are lots of unit tests fortunately.
If you want to be sure everything keeps working you really should add tests. Otherwise you will have to hunt for bugs for a long time after you upgraded your project. And you can't be sure where it goes wrong. In the end it will save you time.