How to get set up FQDN for Azure Container Instance
You can now set a dns-name-label as a property for your container group. Details are shared in this answer - hope this helps and thanks for being a user!
Is there a way to set up fully qualified domain name that i can use instead of changing IP address on every container restart?
Unfortunately, for now, Azure does not support to set a static public IP address for instance, Azure Container Instance still in preview.
In the future, we will expand our networking capabilities to include integration with virtual networks, load balancers, and other core parts of the Azure networking infrastructure.
More information about Azure Container Instance network, please refer to this link.
As a workaround, we can deploy a VM and run docker on it, set static public IP address for this VM, then restart docker we will not lose this public IP address.