You will need to be connected to your home network from the office using a VPN to actually join this machine to the domain and and promote it to a DC.
After that, this DC will need to regularly replicate to the DC at your home. In a normal set-up, you would have a permanent link between the two and they can replicate as and when they need to. It seems you aren't able to do this, so you are going to need to set-up a replication schedule to allow the DC's to replicate when a VPN link is available.
Have a read of this article to understand AD replication and scheduling better.