First, you would want to backup your MySQL database by using this the mysqldump tool:
mysqldump -u root -p --all-databases > alldb_backup.sql

Then you would create your new RoR server, copying over your configuration files from your RoR directory using rsync, or another method. Once it’s running, import your MySQL backup:

mysql -u [uname] -p[pass] [db_to_restore] < [backupfile.sql]

Be sure to test to make sure the app is running well, and as expected before you modify DNS data or shut off your old app server.