As seen in Windows with its automatic updates one can perform such action for Ubuntu as well.
Simply install the corresponding package
sudo apt-get install unattended-upgrades
Here we see how it works
dpkg -L unattended-upgrades
It's done by the cron job an some setting in
Here on can configure what updates are safe to become installed automatically.
One can also setup a email address in this config file for informations when updates were driven.
Did it work for you. Just drop a comment.