Hi Johannes,
In the Azure portal, please check if you have an existing public IP address for your previous VM, using link below:
If the public IP address for your old VM is there, what you can do is click on the public IP address for your new VM, click Dissociate, then when that is finished saving go back and click on the old public IP address, click Associate, choose Network Interface, and select the network interface for your new VM.
Essentially what you are doing in above paragraph is removing the new public IP address from the network interface of your VM and then linking the old public IP address to it instead. Note that this will affect any connections you may have to your VM so if you have something important going on you may want to wait or you could complete while VM is Stopped.
Please let me know your findings/results, whether positive or negative.
Thanks.
-TP