The main thing to do in Tokyo in the evening is eating and drinking, and Ebisu (the neighborhood where the Westin is located) has plenty of both - it's one of the best dining neighborhoods in the world. There are good restaurants on the 38th/39th floors of Yebisu Garden Place, right across the street from the hotel, and hundreds more within a ten-minute walk.

If you want more activities like department-store shopping (until 9pm), clubs, etc., then Shibuya (one stop away on the train or a short taxi ride) has all that. A Shibuya hotel such as Shibuya Hotel Excel Tokyu in Mark City would have more lively street life immediately outside the hotel, if that's what you're looking for.