Hi Potte,
You will first need to create the DNS zone in Azure DNS and make sure you have all your records like in GoDaddy.
Once you create the Azure DNS zone it will show what Azure DNS servers are used. You can then go to Godaddy and enter those nameservers as the new nameservers for your domain.
You mention the ip address 168.63.129.16 this is the so called Azure Magic IP and cannot be used for this functionality as it as only reachable from within Azure.
Hope you get it working.
Rex