Remote upgrading a PC to Ubuntu 16.04 over SSH is a little risky, so it should only be considered when you accept the full risk of having to go physically to the PC and reinstall everything, possibly losing the files. You will need an open firewall for the backup temporary SSH server enabled by the Ubuntu upgrade (port 1022?); perhaps open all ports temporarily to the IP address of your local laptop from the remote PC.
The commands to be run on the remote PC are
screen do-release-upgrade -d