Is there a way to do in-place update the OS of an Azure VM, from within the Azure Portal?
i.e. from Windows Server 2012 R2 to Windows Server 2016
Nope. As per official article:
Microsoft does not support an upgrade of the operating system of a Microsoft Azure virtual machine. Instead, you should create a new Azure virtual machine that is running the supported version of the operating system that is required and then migrate the workload.
Instead, use the Windows Server Migration Tools to assist you in roles and features migration: