It depends a lot on what you want to do. There are many ways to do backups, from plain file copies using
scp to much more sophisticated solutions. If you want to permanently keep the backups, I heartily recommend bup which you can find out about if you google
This solves the "stores backup on remote destination via ssh" and "compresses the backup".
Whatever program you end up using, you can use cron to schedule hourly backups.
In addition the question on how to run a command on shutdown with Ubuntu has already been answered.