The bus terminal in Shinjuku is located outside the west exit, across from the Yodobashi camera 本店, and it's called the 京王高速バスターミナル (keio kousoku bus terminal). Arrive at least 20 minutes before your bus is scheduled to leave, and even earlier if you intend to pick up food at the nearby conbini.
We started at 9pm (from the 6th station, about a 20 minute walk from the 5th station but not much higher altitude) and aside from the usual rest stops (I suggest you take them frequently), we spent an extra hour at the 8th station, since there is little beyond that point, and it was so cold we didn't want to wait an extra hour at the summit for the sun to rise. 10pm is the absolute latest you should start. The sun rose at 4:20-4:30 the day that I went. Don't take your chances unless if you don't care about the sunrise.
Definitely food and water are musts, as the markup is ridiculous on the mountain (though considering how cold it is, you'll be surprised to find yourself buying a cup noodles for 600 yen). I didn't get the stick and I don't think I would. Although it might come in handy at points, I never found myself regretting my decision not to buy one. The nice thing about them is that you can get it branded at each station as you go up, and keep it as a souvenir (they'll even cut it for you so you can fit it in a suitcase). The stick is around 1000 yen, and 200 for each branding.
Oxygen was not necessary for anyone in my group, but your mileage might vary. The air gets pretty thin, but i didn't feel any ill effects.
It's a lot of work, but worth doing at least once. Good luck!