In Finder use Go/Go to Folder to Folder command:
~/Library/Containers
Move the iReal Pro Container folder from your old backup to your new computer.
 
iReal Pro 8.0 and older:
In your backup, look for 2 files using Finder/Go/Go to Folder command:
Type in:
~/Library/Application Support/iReal B
You should see a file called UserSongs.plist.
Make a copy of that file to your Desktop.

Afte that, type in:
~/Library/Preferences and look for com.massimobiolcati.irealbookmac.plist
Make a copy of it as well to your Desktop.

After that, you need to repeat the first step:
Look for 2 files using Finder/Go/Go to Folder command:
Type in:
~/Library/Application Support/iReal B
and drag and drop the file 'UserSongs.plist' from your Desktop.

After that, type in:
~/Library/Preferences and drag and drop the file com.massimobiolcati.irealbookmac.plist
 
Apple support reference for Time machine, Spotlight: (or search for similar)
https://support.apple.com/en-us/guide/mac-help/mh15136/mac