In 2011, I got a bus from the New York Library to Staten Island. They do request that you book your a time, but they werent checking people when they turned up other than for a bib number, and everybody should have one of those!
You get there, queue for the first available bus, and off you go.
Fort Wadsworth (the start) is a cold place. Buy a cheap old jumper that you dont mind discarding at the start, and wrap yourself up! You'll probably be waiting around for a long time over there!
I was told by the organisers that overseas ballot entries were given priority over Japanese residents and that the chance of being successful in the ballot was about 90% (to encourage tourism and the obvious economic benefits that come with it), BUT that was before it had become a major and probably wasnt on many runners radars. I would imagine the interest from overseas runners has probably increased hugely with the prestige of being in that select group of races, and that percentage has probably reduced somewhat. I'd still say you have a reasonable chance though, so its worth a try. Its a brilliant race (I did it every year from 2010 and 2013) and Tokyo really gets behind it!
I did it last year, and doing it again this year (off at 4:10pm on Saturday afternoon). Driving is actually remarkably easy until you get close to the venue, and even then, its not congestion that causes problems, but road closures. Its not particularly well sign-posted around the area, but if you take the A13 and then the A112, it'll lead you straight in to the East car parks. Driving is definitely the easiest way though!
It involves getting to Epcot early, doing the attractions and rides in the morning/early afternoon, then hitting Canada at about 3-4pm for a pint of Moosehead, then Britain for a pint of p*ssy lager, then France for a glass of wine, and onward through the 11 countries in the World Showcase, trying a different alcoholic beverage in each one. It took us a good 5 hours last time, and I dont remember what was going on by the time we reached Mexico, except that we started with 2 people and finished with about 40! Then its fireworks and a bus back to the hotel. And a horrendous hangover!