Yuck! The shuttle experience from the Park Hyatt was a completely different experience! Almost worth walking over there

If you are going to/from anywhere other than the Shinjuku station itself, use the subway stop right under the Hyatt Regency. It does require potentially making a transfer somewhere and only for those that learn the way the Tokyo system works (with the various lines that require separate tickets). However, once you get the hang of it, the convenience of popping out there is just great! We would walk to it from the Park Hyatt as well instead of taking the shuttle.