Only sleeper passengers, and only in their own accomodations. You can get (mediocre and relatively expensive) booze in the diner or cafe car and drink it in those respective cars, but if it’s your own stock, you can only consume it in your room.
Unfortunately, the train will only be in the vast majority of stations for a couple minutes, and you won’t be allowed to get off altogether. However there are occasional fresh air stops (typically where they’re refueling or swapping out the crew), at which the train may be in the station for as long as a half an hour or 45 minutes. But even at those, I highly recommend you stay on the platform right by the train. It will leave without you if you are not back on in time, and it’s always possible for someone to think they know the scheduled departure time, but actually are off by 10 minutes and get stranded. Point is, don’t run the risk, buy your booze in advance, and enjoy the ride.