I don't know that I'd use ODBC. I'd rather go for as direct a connection as possible. You should be able to make a direct connection if it's Microsoft Azure SQL Database or if it's a VM on Azure.

To look at the structures, etc., you can use SSMS. Whatever connection string you have in ODBC should translate well. Once in there, from a database stand point, MASD is just a database for the most part. Clustered indexes, nonclustered indexes, etc. If you are hitting resource issues, you can pay for more resources. It's really hard to make a whole lot more suggestions without knowing what it is that we're talking about, hosted VMs, MASD, Amazon, something else? Each has different approaches to resolving issues. Each has different issues that need to be resolved.

All cloud access from your client is likely to be slower than having metal there on site with you.