Having recently stayed at the Hotel New York, I can put you all out of your misery!
On checking in, we recieved a fastpass "voucher" (a lot bigger than a normal FP ticket) that entitled us to ride any fastpass ride once between 2pm and 6pm - I assume this time window is because most FP's traditionally are taken in the early hours of the day and they wish to spread the load a little better.
You get one FP per party member (so, two then for us) and you only get it for the first three days of your stay - despite staying for 5 days, we still only got six between us. I wonder how many you get if you're staying less than three days?
The passes are dated with each having to be used on a certain day, so you can't use them all on the first day, but the name on it is just the name of whoever booked the hotel, so one person could use all the passes of their group if they so wanted.
We didn't realise they were dated until we got back to the hotel on the first night and inspected them closer (arrived at 3:30pm, checked in to our hotel and rushed to the park as it was closing at 8pm and we wanted to get some rides done). This meant we never got to use our first day's passes and had to throw them away. Which is kind of ironic because that afternoon was the busiest we saw the park and was the day we'd have most needed them - in our four hours in the park we probably only got to do five attractions!
But the other two days we used them both times to ride Big Thunder, mainly cause that had the longest queues. All the other FP rides either weren't long waits, or still had normal FP's available.