It appears to be working if you do not format the drive you install mint on upon installing and enter the same username you used on Ubuntu. However to make sure everything works as you want it to you might set up a system model in a virtualbox to simulate the process before you actually do it on your machine (I recommend that).
In the installation, select Something else on the Installation Type window and set the partition up so you don't format it. Later, enter the username you use on Ubuntu when you set up the first system user.
I tried this on a virtual machine, so no guarantee that anything will work for you - and of course, only the home folder was kept in my test. All installed programs etc. have to be reinstalled.
Before you perform anything on your machine, make backups of all important data.