In theory Google backup/restore should work (although Google does not guarantee it will work with all devices.)
On a OSv4 device I have a setting for enabling Backup and Restore. I have also have tested OSv2.2 (did not find a specific setting for it) but it seems to work.
What is meant to happen is that in setting up your new device, entering in your google account etc. it should restore your apps, settings and data (meaning the songs come in automatically through the cloud.) Or did you set it up manually?

Export your own songs (a single playlist can have up to 300 songs.) Songs and playlists from the forums are still available so you can get them later. This gives you a backup and you can import from there. I would run a test playlist backup email to make sure you can send and receive successfully before taking any songs off the old phone.

Your old phone might allow transferring iReal b to the SD card (in Applications) but make sure you have exported your songs first just to be sure you have a backup - did you try this? I am not sure this would work.