i'm reading about doing the restore to azure sql using azure management studio using T-SQL queries:
https://msdn.microsoft.com/en-US/library/jj852091.aspx
RESTORE DATABASE AdventureWorks2012
FROM URL = 'https://mystorageaccount.blob.core.windows.net/privatecontainertest/AdventureWorks2012.bak'
WITH CREDENTIAL = 'mycredential';
, STATS = 5 – use this to see monitor the progress
GO
Is there a way to do this programmatically using an azure API?
Yes, you can use the Azure REST API to restore. The following link shows the ways to restore a deleted database:
Restore Deleted Database Tutorial through REST
You can build and send a POST request to the following URL with your subscription ID and server name.
Cheers, Luis