On Azure, how do I run a short-lived Docker container on a schedule?
Azure Container Instances
 (ACI) may be a good option as you suggest.  These let you run a container directly on Azure, without having to manage a VM, with per-second billing for the time the container is used.  

Although one of the demos on that blog mentions Kubernetes, the idea of ACI is that you can create a container through the Azure CLI with az container create, just like on your local workstation with docker create.

To create the container, you can use Azure CLI (az command, see quick start docs) or Azure Cloud Shell.

You would need to create/run the container on a schedule from somewhere else - Azure Functions might be a good place to run the "container create" command from  a scheduled function.  This supports bash, PowerShell, and other languages - all running on Windows.

If you want to keep using Docker containers without running VMs or learning Kubernetes, this might be a good option.  

Alternatively, you could move all your code into Azure Functions, but that's a bigger decision.

Update: Jan 2019 - Azure Logic Apps can be used to run scheduled tasks as well.