If it's a free app you don't need root to back it up. Lots of file explorers do this, I use ES file explorer (free no ads)t. To do it with that, inside ES is app manager open that, then long press the app you want, click backup and it will put an apk in sdcard/backups/apps.
Get the apk over to your other phone somehow and click on it inside a file explorer to install.
Since you like the old version of whatever, manually installing from apk also gives you the benefit of the market not trying to update it anymore. (on your original phone you can uninstall, then reinstall from the apk for the same effect if you're keeping that phone around)