Use a cron job with tar and mysqldump to back up and dump everything nightly. If space is an issue, use a tool like rsync to store incremental backups on a remote storage location.
Once you set it up, it's entirely automated every single night and you can certainly do it for multiple servers (rsync can handle numerous servers, but best to set up keys for it).
Similarly with upgrades. You can store all your plugins and themes on a single location and then just use rsync and wget / cron to download a fresh copy of wordpress each night, and push it out to your existing sites, regardless of server. Same with plugins and themes. This will ensure you always have the freshest files each day.
To that end, you'd either need ssh access, the ability to create cronjobs and, ideally, a bash file that can execute all the stuff you want to do via cronjob, or you'd need to ask your host to set it up for you. It's quite straightforward.
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