You can answer that question, if your unattended system with no users logged on has an internet connection, then Yes, if not then No.
If you have a broadband connection it is possible that the connection is established shortly after boot, some require you to connect, that is up to your ISP I guess.
You could test it by setting the VPS updates to Manual for a few days, during that period monitor the
avast! VPS Updates History page.
Once you see a new update has been released then disconnect from the internet, set the Program Settings, (Basic) VPS updates to automatic again.
Reboot your system but don't log on leave it for some time and then logon and see if the update has been installed (about avast). Normally avast checks for updates once you connect or after four hours if you remain connected, so you shouldn't have to wait long.