Do you know if there is any way on azure portal to choose OS family set to Windows Server 2016 with installed .net 4.6.1 version already?
Yes, "Image type" can be changed to Marketplace and you can select 2016 Datacenter.
Does anybody know when there will be any os with .net 4.6.1 available?
Now it is available for Marketplace(IaaS) under 2016 Datacenter. This batch team is currently working to support OS Family 5 as guest Operating system for cloud services.
or should I go with the StartTask feature and try to install .net 4.6.1 manually? However, that option sounds like some workaround and not the correct solution.
You can install it as part of a start task and is considered as the solution for Azure Batch for computing nodes with Windows Server < 2016.