The New City hotel is not in between the Shinjuku Hilton and Park Hyatt, it's actually a bit farther away from Shinjuku station than the Park Hyatt is. However the park in front is totally safe -- I've walked through it alone many times after midnight and never had a problem. Although there are often some homeless sleeping in the park, they never bother anyone. The hotel does run an occasional shuttle to and from Shinjuku station during the day, I've seen it, but if you had to walk, I'd say 20 minutes each way.

From Hilton Tokyo bay to Shinjuku station, best route is Keiyo line to shin-Kiba then transfer to the Rinkai line to Shinjuku. A bit circuitous in distance and about Y200 more expensive, but faster and more convenient, avoids the hike at Tokyo station. Still probably takes at least 50 minutes though if connections work well.