Ride the Shinkansen. This ultramodern and fast eel-like train will take you almost anywhere in Japan, with stewardesses onboard like in airplanes (try the extra sweet iced coffee!). Booking a ticket can be tricky, as sales offices’ employees won’t necessarily speak (understandable) english and the vending machines won’t clearly inform you need both a fare ticket and a seat reservation (on separate coupons) to board the train… If you travel coach, be aware of the 2-3 seat configuration. Also note that rail passes are not accepted on the fastest trains (mainly known as Nozomi trains).
Chase Mount Fuji. Easy? Nope. The (very) shy mountain is often surrounded by thick clouds and weather changes a lot around there… Even on a day with clear blue skies there is no guarantee of seeing it. So check weather forecasts until the last minute. The easiest way to see it is from the Shinkansen train (between Tokyo and Kyoto for example). If you are travelling westbound, seat on the right side of the train (and on the left side if you are heading east). The moment may only last a few seconds before the clouds cover it again, but it really is worth a try.
In terms of dining, Tokyo has a lot to offer. If you want to experience high-end Japanese cuisine, one strong option to consider is Nihonryori Ryugin run by chef Seiji Yamamoto. This upscale restaurant is nicely hidden in the residential side of trendy district Roppongi and awarded with the maximum 3 Michelin stars.
While in Japan, you must spend a night in the serene atmosphere of a ryokan. Really. It’s an amazing Japanese experience. There are a lot of them spread across the country (even in central Tokyo) and they come in a vast array of price categories. The experience is more enjoyable if you go on the country-side though, like the region of Hakone close to Mount Fuji (and fairly close to Tokyo as well) for example.