1. Bullet trains are known as the "SHINKANSEN" if you have the Japan Railpass you can only use the Hikari/Kodama Shinkansen, if you are just purchasing the tickets instead you have the option of using the more frequent NOZOMI Shinkansen.

The shinkansen's depart from SHIN-OSAKA station not Osaka Station, to get from Osaka Station to Shin-Osaka station you can take local/rapid trains or the municipal subway line to the Shinkansen station.

The Park Hyatt is located in Shinjuku, Tokyo, no shinkansens stop at Shinjuku, what I would do and depending on how much luggage I have is to ride the Shinkansen to Tokyo station (every other shinkansen's stop at Shinagawa as well), from Tokyo Station take the Chuo line to Shinjuku Station (15minute ride) at Shinjuku Station I think the park hyatt has a bus service (might want to direct that question to the Hyatt forum).