In-place upgrade of Windows 2016 Azure VMs to Windows 2019 is not officially supported but still something we occasionally need to do. While I’d recommend you spin up a new 2019 VM and migrate your workload if at all possible, it’s a bit long winded but you can do an in-place upgrade.
If you’re lucky it’s as simple as copying the files off the ISO and running through the upgrade wizard, however if it brings up any prompts or messages you need to connect to the console to view you’d not get very far with a service like Azure where you cannot view the console, and this is one of the reasons why it is unsupported directly on Azure.
I’ve done two upgrades so far, one the following way and one just running the ISO. Both methods have worked out fine for me.
First of all you will need access to Azure with permission to manage the Virtual Machine in question, access to a storage account (or permission to create one), a local system running Hyper-V (this can just be a powerful PC running Windows 10), the Server 2019 ISO (or other installation source) and, if you don’t want a very long wait, a decent Internet connection.