Mapping an ACI-instance to a Reserved IP on Azure
Currently, your best bet is to create a DNS A record for the app and remap it to the new IP when you create a new container. However, we are working on enabling you to simply update the container and keep the IP.
Update: You can now request a DNS entry for your ACI container in the Microsoft-managed azurecontainer.io domain using the dns-name-label switch in the az cli. You can keep this DNS stable across updates.
az container create -n helloworld --image microsoft/aci-helloworld / -g myResourceGroup --dns-name-label mycontainer