If I'm reading your question correctly..........I have good news.
A wireless router does not need to have any wired connections in order to operate. You can run all connections wireless.
As you indicated, you must have/install a wireless card in your computer. If you don't feel right about opening your computer, you can use a wireless adaptor that plugs into a USB port on your computer. You will also need the WEP key to make the initial connection. The WEP key is a code written on a label attached to the underside of your router.