There is no other way I think. He should offer you the updates successively, and it should be okay. But for security, personally, I always make a #backup of the directory and the database just before. It is quickly realized and like that in case of problem, I can #restore easily.
For fast backup of the DB:
I create a database in phpmyadmin, for example "saveuna".
Then position yourself on una and go to tab "Operation" -> "Copy database to" -> "saveuna". (The target database must be emptied before)
In case of problems just do the opposite. (The target database must be emptied before)

I know we can use export / import, but copying is much faster.
For the backup of the directory:
I create a directory at the root, for example /saveuna/
Then I use "rsync" in ssh with the command:

rsync -arv --delete --stats  /var/www/una/ /var/www/saveuna/

In case of problems just do the opposite

rsync -arv --delete --stats  /var/www/saveuna/ /var/www/una/

Rsync synchronizes directories source-> target or target-> source and also copies permissions, it's very fast.
I always do this before an update and / or alterations important, fast and safe, if you do not get into the commands ;-)