Although I have not been to the Conrad, I do know the area. It is a bit of a walk to Shimbashi station. Although Shiodome is a new area, I think it can be quite dead at night. Shimbashi station is even worse. On the other hand, the Hilton in Shinjuku is also far from the station and in a business area.

I think this comes down to why you are staying at these hotels. If you are on points, I would choose the Conrad as it is quite new and rooms costs quite a bit more than the Hilton. If I were paying, then I would go for the Hilton, as I do not require the "extras".

As Richard mentioned above, the transportation system is excellent so you should be a quick subway or train ride to most of the places you would want to go in the city.