How to set-up custom domain name in Azure Container Instance
As for as I know, ACI exposes the DNS name using a dns-name-label in azure's namespace. The FQDN like customlabel.azureregion.azurecontainer.io. is provided by Azure DNS service. Unfortunately, there is no way to directly set the custom domain for ACI, but you could create a CNAME record in your DNS provider to redirect your subdomain like www.example.com to this FQDN.If so, you can access your ACI via subdomain www.example.com.
You may consider creating a web app for the container. Then you can create a custom domain for the web app.
cname won`t work since docker sees that the ssl cert is from *.azurecr.io and not example.expemple.com and then proceeds to do nothing. it also doenst work with cloudflare ssledit: i just did more research and i found something very very promisinghttps://github.com/Azure/acr/tree/main/docs/custom-domain
cheers,Zakaria