RichardInSF

I timed the walk a few times from the hotel to Shinjuku station south entrance on my last trip and it was 12 minutes for me. I've rarely had the cab cost more than the flag drop of Y710, the easiest and quickest is to have the cab take you to the west entrance, lower level. If you are going south on the Yamanote line, take the cab to Yoyogi station, which is also Y710, make sure the cab turns left leaving the hotel.

The subway system, TOEI Oedo line, is about 8 minutes walk to Tochomae station or about the same if you walk to Hatsudai station at the Opera building on the Keio New line (which becomes the TOEI Shinjuku line), reached by walking in the direction away from Shinjuku. A third option is to walk towards the station but get on the TOEI Shinjuku line which, if you pick your entrance properly, is also only about 8 minutes walk.

There is also the 100 yen bus which leaves from stand 20 at the west exit from Shinjuku station (go up to street level and turn left about 150m). Get the bus labeled "West Side" which leaves about 6 times an hour until 7 or 8 pm. It drops right at the hotel. You can also take it from the hotel stop but only up to 10am and only on weekdays as otherwise the route to the station is a bit circuitous.

If that's not enough choices, let me know and I'll try to come up with some more.