It's a cliched shot, but still one that is nice to finally have in the bag! Given that it is a little awkward to reach the right spot, I thought it would be nice to put together a quick guide to getting the famous "Bullet Train whizzing past Mount Fuji" pic.
It all starts with getting to the right station. Yoshiwara on the Tokkaido mainline is where you get off for the shot. To reach this point from Shinjuku station in Tokyo requires two changes: one at Yokohama, then again at Atami. It costs around 4200 yen and takes around 2hrs 15 mins. To check up to date train times, check Hyperdia.
From Yoshiwara station, it is either a 30-40 minute walk or a short taxi ride to the photo spot. When we went there seemed to only be two taxi drivers in the whole town, and provided you are holding a camera and can say "Fuji" and "Shinkansen" they know where to take you. The spot is in a rice paddy to the south of the Tokkaido Shinkansen line. See the map below for geotag of my location. The exact GPS coordinates are 35°9'5" N 138°43'29" E.
Once dropped off at the field, experiment with different compositions and locations. It's surprisingly tricky to get an uncluttered shot. Japan has amazing infrastructure but doesn't do a good job of hiding it! There are signs and clutter in the background. If possible avoid the bridge (shooing from the right of this GPS location). Get as low as possible in the field to avoid getting too much of the background in: i.e. so that you just have Mount Fuji as the backdrop.
If you arrive with a decent amount of time, you should have plenty of chances to get the composition and settings right before the light gets right. Trains zip past every few minutes and you can hear them coming.
I found a high shutter speed was needed to freeze the bullet train (>1/640), which meant a loss on depth of field in the foreground and ramping up the ISO. I also used high speed burst mode so that I could have a higher chance of catching the train. Once you get back and review the photos you'll be amazed at how easy it is to get the train in an off-balance position.