This feature has been available in Azure for a couple of years now, it's as simple as ticking a box and setting a time for shutdown. You can also have it fire a webhook if you need any notifications, or scripts run, or anything else when it shuts down.
Also on Azure there are pre-configured data science VMs and deep learning VMs with all the tools you need including R already installed, you just pick a size and spin one up. You can go small just to try the software, or go for one with multiple GPUs, all already set up with the tools and usable straight away. You can even reboot the same box in different sizes depending on what you need that day!
You are correct that while a VM is not running, you will be charged only for the storage used.