If you purchased iReal Pro in the Mac App Store updates should appear there.

Open the Mac App Store from the Apple menu in the top left corner.

In the App Store make sure you are signed in with the same email as you used to purchase iReal Pro:

Click on "View My Account" and iReal Pro will be listed there with a button telling you if an update is available.

If you don't see iReal Pro listed you are probably logged in with the wrong Apple ID account.

NOTE: on some older systems since we changed the name of the app from iReal b to iReal Pro you might need to go in the Applications folder, delete the old iReal b and then update the icon in the dock to the new one by dragging it there.