Yes that’s how a lot of people connect their 360’s to the internet, put one end of the cable in to the 360 and the other in to the router. Then sign up for Xbox Live
You might have to alter a few settings to get an open connection to xbox live but lets worry about that once your connected.
Yeah thats how i connect to live , iam using a Linksys router .
Works very good, you can also connect to a laptop and get on live .( but i had problems doing it that way).