Should I just direct my domain name to the public IP address?
As Mohit said, we can set static public IP via Azure portal, and map your domian name to that Public IP address.
Because the IP is dynamic, if the port is destroy and recreate again,
a new IP is generated.
By default, AKS expose pods to internet will create a Kubernetes service, the Public IP address work for that service.
If one pod was not work(multiple pods), AKS will create another pod in your service and that will not get a new public IP. But if you only have one pod in that service and re-create that pod, we will get a new Public IP address.
For now, Azure does not support to keep the public IP address for AKS service.
Hope this helps.