I have been struggling to find some resources that help explain, how we use the File Storage with Web App Service.
There are ways to use it with the old Web Roles, Check here (Using the Azure File Service in your Cloud Services (Web and Worker Roles)).
However, there is NO OnStart() methods in "Azure Web Service".
If you're looking for mapping a drive to your Azure File Service share in your Web App using SMB protocol, then I don't think you can do as of today.
What you could do is make use of Azure File Service REST API and manipulate shares and files that way. You don't have to actually use the REST API per se; you can simply use the latest version of .Net Storage Client library which is a wrapper over REST API.
Please see this link for more details on how to use this using Storage Client library: https://azure.microsoft.com/en-in/documentation/articles/storage-dotnet-how-to-use-files/.
