I played the black course yesterday. Living out of state (CT) I was only able to make a reservation 2 days in advance. On Friday night, I called at 7pm to reserve a twosome. After about a dozen attempts through the system I successfully booked a 10am time on a Sunday morning.
We checked into at 8:30am, and I noticed there was a foursome completely open at 11:21am. Mind you, it was partly cloudy yesterday, 80-85 degrees on a Sunday in late summer. I was shocked there was an opening for a fourball at a non-twilight hour.
While you probably do not need a reservation, walking on as a single may require a long wait. I think it's fairly hit or miss based on no-shows or cancellations. I recommend getting your name in the reservation system and booking a single. You will probably be successful walking on, but you will have a better day if you aren't sitting around for hours waiting to tee off.
The course is closed on Mondays. The message on the reservation system gives details regarding events, course closures, aeration schedules, etc. 516-249-0707.
While the course was incredible and humbling, be aware the round will be over 5 hours. Yesterday we played in 5.5 hours while waiting on nearly every shot. It wasn't just the group in front of us, the course was backed up everywhere.