The best solution would be to run a standalone DNS and have it on always. My larger IT department does this with some kind of linux/unix DNS servers (and DHCP). I'm not sure what kind of work they need to do with the AD and Azure, might be as simple as having AD update this standalone DNS, another config option when building your AD. Windows AD does not require Windows DNS, but it does require a DNS server it can modify.
That said, it's much easier to just blast through the vanilla AD install and let it handle the Windows DNS (nice and integrated), which is I'm guessing what the OP did when setting up the DC.
If I was the OP, and I wanted a DC running, I'd buy a "cheap" little pc and just run it. I have Zentyal CE running at home on a Mele Quieter2 which is a very low power device. It works, and it works fine for DNS, DHCP, etc. If the goal is to learn Windows Server, then buy this small PC and run Zentyal CE on it, join it as a second DC to the Windows domain and make it the primary DNS and DHCP server so it can run all the time.
That said, it's much easier to just blast through the vanilla AD install and let it handle the Windows DNS (nice and integrated), which is I'm guessing what the OP did when setting up the DC.
If I was the OP, and I wanted a DC running, I'd buy a "cheap" little pc and just run it. I have Zentyal CE running at home on a Mele Quieter2 which is a very low power device. It works, and it works fine for DNS, DHCP, etc. If the goal is to learn Windows Server, then buy this small PC and run Zentyal CE on it, join it as a second DC to the Windows domain and make it the primary DNS and DHCP server so it can run all the time.