From Shinjuku Station to Nihonbashi Station:
Take the Marunouchi Line to Akasakamitsuke Station, change trains to the Ginza Line to Nihonbashi Station (19 minutes, ¥200)
From Tokyo Station to Nihonbashi:
From Tokyo Station walk to the exit marked Nihonbashi Exit, this will take you out directly into the Nihonbashi area.