Apparently the command "do-release-upgrade" is not advisable over SSH:
Continue running under SSH?
This session appears to be running under ssh. It is not recommended
to perform a upgrade over ssh currently because in case of failure it
is harder to recover.
If you continue, an additional ssh daemon will be started at port
'1022'.
Do you want to continue?
That warning is simply letting you know that if you update SSH while you are connected, you might lose your connection. If that happens, a second ssh session has been opened on another port that you can connect to using:
ssh -p 1022 user@your.ip.address
Additional, the console in the DigitalOcean control panel connects to the droplet directly. If you lose connectivity over the network, you can always fall back to the console.
Yes, but is it suggesting that there is some other way of upgrading Ubuntu--asides from physically accessing the server?
It's suggesting that having physical access is the only completely safe way to upgrade. The console in the DigitalOcean control panel mimics having physical access.