Unattended upgrades

'unattended upgrades' is the name of the package on Ubuntu that will give you automatic security updates

• Desktop users
• Server Guide

The Server Guide suggests installing apticron to get notification emails

If you're running at least 16.04, you can also sign up for the LivePatch service (free for 3 machines) which can patch the kernel without reboots. Similar services from RedHat etc. cost ~2K per year.