The Hilton is near Shinjuku station. When it gets dark, ask at the hotel how to get to the "Oedo" subway line and take it to Roppongi (11 mins). Go up several long escalators and follow the signs to Roppongi crossing -- it's the most foreigner oriented night life area in Tokyo. Wander around, have a drink at a decent looking bar, etc.

Earlier than that, wander through the camera/electronics stores near Shinjuku station (and about 6 minutes walk from the Hilton), and also check out "Tokyu Hands" department store in Takashimaya Times Square on the opposite side of Shinjuku station.