How to link my Azure VM running SQL server to my on-premise domain or Active Directory?
Yes you can.  You'll need to have a site to site VPN between Azure and your environment.

Once that's one you've got a couple of options.


You can change the DNS settings on the vNet to point to your office domain controllers.
Add domain controllers to Azure (with static IPs), then change the vNet to use the Azure DCs as for authentication.


Once one of those is done, reboot your VM and you should be able to add it to your domain. We do this for clients all the time, and we've done it with our 
lab, and it works perfectly.

The product Azure Active Directory isn't needed to do this at all.