Yes, but ...
Two possible problems -
Most wireless routers need a cabled connection to setup the wireless ! Once it's configured, you don't need the cable.
Not sure if it's still the case, but with an STB you used to have to
register each new device. (may not be neccy now)
So, yes he can have a wireless router with single PC, but might need to move the PC into the lounge to set it up (or plug in a friends laptop.