On a mostly cloudy afternoon, I found myself making the short trek out to the Tokyo Metropolitan Government Building, a two-tower monstrosity in Shinjuku's Skyscraper District.
Basically, head straight west out of the station and down the large pedestrian passageway for about a kilometer. Eventually, you see the two-pronged plug sticking into the outlet called the sky.
The Government Building is completely free and usually open 9AM-11PM, though one tower or the other might be closed on a Monday or Tuesday. The South tower closes at 5:30PM if both towers are open, and both towers are closed around New Years Day (but open on January 1). Through November 2013, the elevators are under maintenance and the lines might be longer due to less service, but if you visit on a weekday when school is in session you should be fine. Exit to the west from Shinjuku Station and follow the signs to Skyscraper City and the building.
Kabukicho can be found by following the exit signs from Shinjuku Station (towards the northeast), and then following the crowds. You can visit anytime, though the nightlife kicks off around 6PM.