The best solution we could come up for this issue with was to create a Private DNS zone, give the container's service principal or identity Private DNS Zone Contributor access, and run the following as part of the entrypoint script for the container:
az network private-dns record-set a update --name <name> -g <resource-group> -z <zone-name> --set aRecords.ipv4Address=$(hostname -i)
I'd imagine a similar solution will work using other APIs or with public DNS Zones.
This is the only way that currently seems to handle the rotation of the the IPs on container startup.
Note, I'd really like to see similar functionality built into the Container Instance offering to make them more functional out of the box.