Can I get the older reminders app back?

Essentially, no. There was a warning and an option not to upgrade Reminders when installing iOS13.

Get ready for the new Reminders app - https://support.apple.com/HT210220

I have not used these steps so if you use them they're at your own risk.

The Reminders app can be deleted and re-downloaded from the App Store.

• 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

Again, this may give you the Reminders app before it was upgraded, but your reminders data may have been upgraded and won't work with the non-upgraded app and you may have to re-enter all your reminders.

Yes, you can get the old app back but your actual reminders have been upgraded and won't work with the old app.

You would have to take a picture of all your current reminders, delete them from iCloud.com and manually reenter them after adding the old app back. If you were successful adding the old app back you would need to remember to read the warning and choose Upgrade Later when you open Reminders.

How do you get the old app back?

I think it would just download the version of the app for the iOS version on the phone, namely 13. Nor can you downgrade iOS.