A air-gapped computer should NEVER be exposed to the internet. It should be behind a firewalled network, not even a normal local area network over a router. It shouldn't be connected to any network, where any PC is connected to an unsecure network, or the internet.
An example; Enterprise clients at the old datacenter I worked couldn't even setup a server from the internet, they had to call, and it would be set up through the air-gapped system where their sensitive information is stored, then server(s) set up for them on the datacenter network.