Alberto,
There's no way to run Autotune after your subscription expires. The configuration tool is part of your subscription and it's always being updated based on changes in Facebook and the latest release of JFBConnect.
If you want to manually configure your application, you can do so in the
Facebook App Developer
area. In there are all the settings that JFBConnect uses.
The warning messages that you see in the admin area can't be removed without running Autotune, but they shouldn't cause you any issue. If you really want those to go away, you'd need to copy the #__jfbconnect_config table from your 'old' site to your new one. That will import all your previous settings, including the last values stored from Autotune. That may be the best way to go for making sure everything is configured the same, but that's up to you.
I hope that helps explain,
Alex