You can install Mint in the same partition(s) currently occupied by Ubuntu. In the installer, select the "Manual Partiton" option and identify the partition used by Ubuntu as your Mint "/" partition. I'm not sure if there is a "migration assistant" available for Mint to assist in keeping your Ubuntu settings found in "/home" as part of your new Mint (I always do fresh installs on any ops).
I'm not familiar with MyPlaces so I can't comment on doing a backup and restore from Ubuntu to Mint.
At any rate, do a backup of critical Ubuntu data to a removable device or media before moving on to Mint.
Linux'd since 2005 Linux ID 422356 Ubuntu ID 15015