Technically speaking, the GUI version of apt-get update does it automatically already - those already cater to the needs of a desktop user. Command line tools are more of a technical type of user, typically admins , who know what they're doing.
However, there's nothing stopping you from making a script out of it, and reviewing logs every once in a while. For example, here's a quick sketch:
#!/bin/bash
main()
{
local DATE=$(date +%Y_%m_%d_%H_%M)
local LOGFILE=AUTO_UPDATE_$DATE
local DIR="/home/localuser/logs" # where to store logs
apt-get update &> "$DIR"/"$LOGFILE"
}
main
And use that as script to run upon shutdown or reboot using /etc/rc6.d directory scripts or alternatively - cronjob to schedule this script at specific time of day. Remember though that checking logs will be your responsibility.
In future, there will come snappy - a newer system for transactional updates which is now is in very young stage and supposedly should come to 16.04. My experience with it is somewhat limited, but on Raspberry Pi it does update automatically and reboots itself once newer version of packages are available, sort of how Windows update works
Addition
Per muru's suggestion one could use unattended-upgrades to automate the updates as well, and probably in a less verbose way than my solution.