HND to Shiodpome or Conrad is a 20 minute cab ride. If you are lightly packed then a commuter train will work for you also. The station get off is Shiodome SIO-SITE. most of the lines stop there. This will put you exactly under the Conrad. You can also get of at Ginza and take a short 5-7 minute cab ride to the Conrad.

BTW the closest site to the Conrad is the Hamaryku Garden, about 5 minutes walk. It is quite nice but in January I don think there will be much to see. You can catch a local tour boat from there and tour the Tokyo bay. Quite Nice. ^