That's not a path, that is a URI (and a uri-path or url-path is the part after the hostname starting with the slash). When asked for a path to a file or program, you need to specify a filesystem path. Given username "username", and domain "example.com", here are what you need to use for situations:
/home/username = filesystem path to user "home directory"
/home/username/example.com = filesystem path to "web directory"
It is not a good idea put shell scripts in the web directory though. Try making another directory for shell scripts and other files that you don't want people to download from the web site:
/home/username/bin = maintenance programs and scripts
If you do have files in the web directory that shouldn't be downloaded, change the permissions to deny world read access. (ie 750 for directories or 640 for files).
As for setting the cron job, your path might look like /home/username/bin/automysqlbackup.sh - but also don't forget to set the permissions on that file to include execute access.