Yaesu Central Entrance (eastern side) is probably the closest for Shinkansen. But even though Tokyo Station is large, it's sensibly laid out with corridors north to south and good signage. If you end up on the wrong side (east/west, Yaesu/Marunouchi) there are crossover corridors that don't require you to enter the JR station.

I'm presently staying at a hotel right outside the Nihonbashi entrance (northeast side) and I can get to the Chuo JR platform in < 5min.

If you want not to be stressed you can get there a bit early, though not much will be open except for a few convenience stores and kiosks.