If I had any interest in setting up something like this. I would probably combine a cron job (linux built-in scheduled tasks) with an existing cloud sharing service like Dropbox or Google Drive or literally anything else with a folder path.
First I would back-up everything manually, because Linux has fewer guard rails that prevent you from deleting all your files.
Next write a script to copy over the files from the local steam save folders to the linux dropbox folder and save it as a file 'backup.sh' or whatever. I think the main path in Linux for the Dropbox is ~/Dropbox.
It would take some time to tinker to get the correct folder paths at first, but the script to copy the files would look something like this.
cp -rv path_to_source path_to_destination/
Once it works manually (i.e. files are copied from the source to the destination) it is time to automate. The v should show progress.
The next step would be to automate this by running it as a
cron job every day at midnight or every hour or whatever interval works. Every half hour maybe.
30 * * * * /root/backup_to_dropbox.sh
Now that this works I want to prevent copying older saves over newer ones. So I adapt my script to:
yes|cp -ur path_to_source path_to_destination
The -u checks if the file is newer than the destination file, the R includes subfolders.
Now I want to do it the other way around but probably with a different interval like at the top of the hour with another cronjob or simply 5 minutes after every reboot.
0 * * * * /root/load_from_dropbox.sh
@reboot sleep 300 && /root/load_from_dropbox.sh
The copy command should literally take (micro)seconds but dropbox might need some time to sync. But in this case older save files will be overwritten if newer ones are available on the dropbox.
yes|cp -ur path_to_destination path_to_source
In theory the hard part is done, the files are now available as a backup in Dropbox.
For Windows there's probably an add-on or button or whatever to sync with the steam folder.
Even easier might be a symlink, simply create a symlink folder in Linux between the steam save folder and the dropbox folder so Steam saves directly into Dropbox. This is more risky though.
ln -s ~/Dropbox/Steam ~/path_to_steam_saves