So, will Windows Azure work for that
Yes, Windows Azure will work.
or is there any other services that can provide something like that?
If you want to develop and run a .NET game server, you'll need something that will run the .NET framework (as well as any other libraries you end up using in your project). Any server host that offers Windows Servers will suffice.
The two big options for Windows Server hosting are Windows Azure and Amazon EC2. There are others out there as well, some being a tad more reliable than others.
How do you choose your host? Personal preference. Finances. Methods of access. Reliability. Reputation. None of us can answer that for you.
If you have the ability to install the software you need (.NET, any dependency libraries, Visual Studio, SQL Server, whatever else you require to build and run your game server), and you can access the system in a way that works for you (ssh vs. rdp vs. telnet vs. vnc vs. http) then the rest is fairly subjective.
Windows Azure has a 90-day free trial. Give it a shot. If it works, keep it. If not, check out the others.