As I recall, there is an indication on the exit map in Nishi-Shinjuku metro, be aware that exit will be closed with a gate by 9 or 10pm?
If you walk from/to Hilton and Nishi-Shinjuku, you will have to walk that underground path (just on the left side if you are just outside Hilton enterance or take lift on the right side inside Hilton to basement)

not sure about the pass, as I always use their IC card
but I suppose the pass can be bought with the machine by cash