I'm developing ASP.NET Web API services and placing these onto an Azure Service Bus queue to be processed. The Web API services are hosted on Azure.
I need to implement an application that listens for these messages and processes them when they are received.
I'd like this to be hosted on Azure but not sure of the best way to approach this.
There are several things you can do. You could use ASB's OnMessage API which allows you to register your callback and handle incoming messages with concurrency and auto-completion.
On Azure you have several options: Cloud Services (worker roles), Azure Web Jobs, Azure Functions (if your processing is fast, otherwise I'd not recommend it), Service Fabric (might be a bit of an overkill if system is small), and plain VMs if needs to be.
Warning about functions - if you do intense work, Functions are not ideal as you'll pay for time/memory you execute.