When creating a Container Instance, you can specify a DNS name-label value that customizes part of the URL Azure exposes for you for accessing the container (the Fully-Qualified Domain Name or FQDN field in the Container Instance’s Overview Page). Unfortunately, there’s no native way of specifying a custom domain name for a container instance.
However, there are two approaches to this limitation:
Azure Application Gateway is a web traffic load balancer that enables you to manage traffic to your web applications. Traditional load balancers operate at the transport layer (OSI layer 4 - TCP and UDP) and route traffic based on source IP address and port, to a destination IP address and port.
Read more about Application Gateway here
