I read through a good blog post from a colleague Damien Caro on the monitoring of Windows Azure applications with System Center 2012(Operations Manager). Definitely worth reading and trying out, especially if you are using System Center for managing your IT environment. Another great overview is the blog post of Adam Hall.
Basically, after installation, the Monitoring Pack for Windows Azure (in Operations Manager) offers you the following functionality:
On the similar note, I have also looked at the System Center 2012 App Controller, which enables you to manage your applications across both Private (VMM based) and Public (Azure based) clouds. So you can easily deploy and update your applications (even multiple-tier, multiple resources) via a single UI. See more interesting details in this video series.
Finally, I would also recommend a whitepaper by David Chappell on the topic of The Microsoft Private Cloud: A Technology Overview, as it describes private cloud technology in details, links it to System Center 2012 and mentions how it integrates with Windows Azure.
Really good stuff and need to know for every IT professional working with Microsoft technologies.