Hi @Lutz Rahe ,
On the question "I cannot choose Hybrid Group Account", did you mean that you are not able to see your Hybrid Worker Group while trying to start the runbook as explained here?
On the question "How can I create the script or trigger the service to run a script on the VM?", You may save the runbook in the Azure Automation account itself and start the runbook as explained here to run it in the VM environment that's part of the Hybrid Worker Group.
On the question, "How can I add this Hybrid as the target into Azure Monitor", as explained here you may forward Azure Automation diagnostic logs to Azure Monitor.
Let me know if you have further queries with regards to it.