No, I wouldn't call it near the Park Hyatt. On foot it would probably take you half an hour or more. Best to go by taxi or subway, but you'll have to figure out what line to take.
It's right in the middle of the gay district (Shinjuku 2-chome), facing a small park that may just be named Shinjuku ni-chome Koen (I'm not sure).
Actually, there is a Japanese language map on their site, here:
24 Kaikan Shinjuku Branch Map
The two boxes with text and a line pointing to a dot are showing nearby subway stations: Shinjuku 3-chome Station (the one with the 3 visible in it) and Shinjuku Koen Station on the Marunoichi Line (the other one).
Once you're in the area, you should be able to find the sauna when you find the little park in Shinjuku 2-chome. Just look for the sign with "24" on it, or ask any of the lots of obviously gay guys who frequent the area.