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.