I am wanting to do the same thing.
One issue is you have to have a valid SSL cert with the custom domain to support the SFTP protocol. Storage accounts don't support adding certs directly right now and only offer SSL protection for the native domain.
From what I've gathered, the only way to add a cert to a storage account is using a CDN endpoint as described here. I did that and was able to get https working for accessing files over ssl. However, I couldn't get SFTP to work. I think this is due to the port used (22) and maybe the CDN doesn't support SFTP regardless. I did change the https port on the CDN to 22 and it did not help.
@Microsoft, are there any plans to add custom domain support directly to storage accounts and fix the SFTP for custom domain issue or some other way to add custom domains to the new SFTP server?