Find a New York state resident to play with you. They can register their drivers license with Bethpage Park and make tee times through the automated phone system. Don't bother registering as an out-of-stater. I faxed my info in and never heard anything more from there. Besides, NY residents can make tee times further in advance than out-of-staters (7 days v. 3 days, I think), so there won't be tee times left by the time you can call in anyway, most likely.
My brother lives in NYC and we've had success both getting tee times through the automated phone system and by making tee times on other courses at Bethpage (red, yellow) then switching up to the Black if there are open tee times. At the cashier's desk in the clubhouse, they have TV screens that show all of the available tee times for each course that are open for the day. We've walked in to pay for an 8 am tee time on the yellow and walked out with a 11:15 am tee time for the Black. Be flexible and optimistic - it's worked out most times for us.
Unfortunately, once you've got your starting time, you're on your own. The course is a monster, you have to walk it, the rough is brutal, and it's totally beautiful. Enjoy the walk, hit a few memorable shots, and take your lumps!