@Roland Eß Apologies for the delay in response and all the inconvenience caused because of the issue!!!
Assuming from the the heading of the question I do understand that you might want to understand the migration process involved from Azure Cloud Service(classic) to ARM. Azure Cloud Services (extended support) is a new Azure Resource Manager based deployment model for the Azure Cloud Services product. With this change, Azure Cloud Services running on the Azure Service Manager based deployment model have been renamed as Cloud Services (classic) and all new deployments should use Cloud Services (extended support).
You can read about the migration process here.
You would also like to check this article to understand migration in Azure for classic services to ARM.
Now coming to the second part of query which is different from the query topic I do understand that there is issue with VM connectivity which you might be facing on your side I might be wrong so feel free to correct me if I am wrong since there is not much error mentioned as such I would like you to go through this Troubleshooting guide for connectivity related issue on your VM.
If this is not helpful do let me know what exactly is the error so I can help you better. I would suggest you to check the credits remained in your subscription as well which might be another reason that your VM is turned off. Do let me know if none of the above is the use case we can connect with Support as well for further help.
Hope it helps!!!
Please "Accept as Answer" if it helped so it can help other in community looking for help on similar topics.