Should not have updated the reminders app on the iPhone.

I have not tried these steps but I think they might work.

• Backup your phone first
• Turn off Reminders from Settings / iCloud
• Sign out of iCloud
• Delete the Reminders app
• Open the App Store and search for the Reminders app
• Install it
• Sign back into iCloud
• Turn on Reminders
• Open Reminders and check to see if you see the warning about upgrading it - but DON'T UPGRADE. Instead, tap Upgrade Later

If it works, all of your Reminders should sync back and you should then be able to sync with your Mac.

Reminders Warning!

Upgraded reminders aren't compatible with earlier versions of iOS and macOS. If you upgrade your reminders on your iPhone with iOS 13, your iPad and Mac using the same iCloud account can’t access your reminders until iPadOS and macOS 10.15 Catalina are available.

You really should have taken the time to read the warning message you were given when you chose to upgrade the reminders on your phone. Not only did it tell you it wouldn't be compatible with devices running older versions of iOS and macOS, it told you exactly what devices you were signed into that it would not sync with.

Your only real option to get them syncing up again is to upgrade your Mac to Catalina.

Why not just upgrade your Mac? It's free.

Word of advice, the steps I outlined (use at your own risk) might ONLY rollback the Reminders.app but not the reminders data because of iCloud.