You basically move the files into the DropBox directory and they should sync, or you can create symlinks (in the Dropbox directory) to them where they get created.
As for a "monthly" basis, if you mean each month, then you will have to edit a script to move it over, but if it was me I'd simply keep 7 days worth (1 for each day of the week) of the files you have. There are plenty of scripts out there that you can get and modify that will do that for you.
A simple one can be found at https://servinglinux.com/thread/4-backup-script/. In that one, if you are doing multiple DB's and directory backups then you'd want to have multiple copies of it (one for each DB/site) and then call them via a Cron job. You'd also want to modify the
to have the file name prefixed with something like
and replace the _db with _site if you do the actual site files separately. In fact, you can actually remove the $hostname from it if you want. I used it when I had multiple servers online and it allowed me to keep up with which went where.