There is no direct connection from Roppongi to Shinbashi Eki. However, you can take the subway to Hamamatsucho then take the train to Shimbashi. Alternatively, Toei Subway operates a vehicle from Roppongi to Shinbashi Eki every 10 minutes. Tickets cost $1 - $2 and the journey takes 9 min.
How to get from Roppongi to Shinbashi Eki
From Roppongi subway to Hamamatsucho then train to Shimbashi. 10 min ¥295. From Roppongi subway to Shiodome then walk to Shinbashi Eki. 16 min ¥195. From Roppongi taxi to Shinbashi Eki. 2 min ¥2079. From Roppongi Uber to Shinbashi Eki. 2 min ¥3076. From Roppongi walk to Shinbashi Eki. 40 min.