How are you packaging your site in blog storage? One way to publish to Windows Azure Web Sites is to use web deploy. You could package your site as a web deploy package as part of your build, and then use msdeploy.exe to push your bits to the site: http://technet.microsoft.com/en-us/library/dd569106(v=ws.10).aspx
This would be no different from using web deploy to publish to an IIS Server.
It looks like someone else has used this approach with success: http://robdmoore.id.au/blog/2013/06/01/windows-azure-web-sites-programmatic-web-deploy/
Hope this helps!