Microsoft does not have a backup system available for its Websites in Azure.
However you can piece one together with the following two parts and add them to a cron to get a backup done each night
/usr/bin/mysqldump -h <server> -u <username> –password=<password> –single-transaction > sql`date “+\%Y-\%m-\%d”`.sql
wget -r –level=0 –quiet -P `date “+\%Y-\%m-\%d”` ftp://<username (don’t forget to escape the slash)>:<password>@<server>
If you have loads of themes and uploads on the server it may take a few moments to get the copy done. I tested a default install and it took about 10 to 15 seconds. This is an incentive to reduce unnecessary clutter inside your wordpress installation.