Can I get Wi-Fi in a distant room using the Ethernet wall port?

A Wi-Fi access point is made for this. Connect it to the wall port with an Ethernet cable, then configure its wireless network. Devices in that room can connect over Wi-Fi while the access point uses the house wiring to reach the main router.

First, test the wall port with a computer if you can. The cable at the other end needs to be connected to your router or a network switch, and the port needs to be working.

If you have a spare Wi-Fi router, check whether it has an access point mode. That can do the same job without buying a dedicated access point. Follow that router's setup instructions, since which port to use can vary by model.

Access point mode leaves the main router in charge of assigning network addresses. You can set a separate Wi-Fi name for the room, or use the same name and password as the existing network.

Some Wi-Fi extenders and mesh units can also use an Ethernet connection. Check the specifications for a wired access point mode or wired connection, rather than assuming every extender supports it. Pick one that supports the speeds you need.