There is an official Microsoft "Robocopy like command line tool" to copy data from/to Azure Blobs, it's called Azcopy, latest version is available here : http://aka.ms/downloadazcopy
Unfortunately, Azure Blob Storage doesn't allow the creation of a dir structure (there is only one level available called Container) so it's not a good solution for you.
So In your case, I would rather recommend to use Azure File (An SMB 3.0 File Service) and connect a Windows machine (8 and later, 2012 and later) to it. Network Trafic will be encrypted (one of coolest new features of SMB 3.0 : SMB Encryption), Authentication is done using Azure Storage Account name & key.
Few steps to configure :
create a Azure Storage Account
Create an Azure File Share in this Azure Storage Account
On a Windows Server (2012 or later) or Windows Client (8 or later), mount the Azure File Share as a network drive
Use robocopy to copy your data
More information : How to use Azure File