You can do that with unattended-upgrades. But no, of itself Linux Mint does not have support for unattended upgrades, and the default implementation on Linux Mint, and our recommendation, is for you to review available upgrades in the Update Manager before installing them. If you want upgrades to be available less often, you can in the Edit > Preferences menu configure how often Update Manager checks (default is every 15 minutes).
Update Manager also has a set of rules to determine the level of an upgrade (see Edit > Preferences for details on the levels), with by default only level 1, 2 and 3 being offered for installation by the Update Manager. This reduces the risk for new users of becoming stuck with an unbootable system, from applying upgrades that might introduce new problems with their hardware (like kernel and xorg driver upgrades). Using unattended-upgrades is not configurable to the same level of detail, and hence you should probably not use it if you are not comfortable with the terminal and with rolling back package upgrades manually.
Anyway, here is one example of installing and using unattended-upgrades: http://community.linuxmint.com/tutorial/view/1217