first, the instructions you were following were for the auto update and that is not the correct method to use.
since you are upgrading two versions you need to use the full package method found here:
https://www.phpbb.com/support/docs/en/3 ... upgrade31/
follow those instructions exactly and in order. do NOT leave out any of the steps.
as to your VPS. you should have complete access via ftp to your server.
if you do not, then you need to contact your host to find out why.
if you have the complete access to ftp, they you can do the upgrade.
make the database backup from your current phpBB admin panel. save it to your computer.
then use your ftp access to backup and files and then follow the instructions above.
once you have uploaded all the new files there will be a folder named "install" in your root directory where your board
in your browser, go to: yourdomain.com/yourphpbb/install
then you can see the upgrade/update page and follow those instructions above.