How to upgrade Minecraft Paper edition without breaking the world?

Try the upgrade on a copy of the world first. It may not be safe to just revert to the old jar file after the world format has been upgraded.

Step by step

Check Java requirement, plugin support and Paper notes for the new Minecraft version. Keep the jar of the working version.
Shut down the server and backup all worlds, plugin data and configuration. If there is an external database, make a copy of it at the same point.
Open it with the new jar in the test directory. Check world bugs, Nether and End, player inventory and critical plugin commands.
Once it goes live, test it with a small group of players first. If rollback is required, restore the previous world and configuration along with the old jar.

Re-opening a world copy opened in the new version with the old version.