The cheapest way i can think of since you don't have wi-fi is to connect it via a ethernet cord. You can connect it via ethernet temporary and update wii which you can then play the game you want. You don't have to keep it connected if you don't want to.
USB Ethernet Adapter
^^ This is what you need. Once you get that get your ethernet wire and connect to this device, this will go into back usb port of your wii, then you can update your wii!
Once you did that go
for a guide to connect to internet and also free wii points/game while you are connected
or if theres a way to update with USB, that i don't know of.