The Royal Park Shiodome is on the Airport Limousine Bus route, so that's the best option, as long as you can board by around 6:45PM--it depends on when your flight arrives.

Otherwise, take the N'EX to Tokyo Station and a cab to the hotel.

You're within easy walking distance of two subway lines and the JR Yamanote Line, which circles the central city.

As for what to see, that's up to you. Buy one of the three or four city guidebooks to Tokyo that are available in any large bookstore. (There are also city guidebooks for Kyoto.)

I, too, recommend sending your luggage to your Kyoto hotel. Just take a carry-on or daypack for one day's change of clothing. The price for each bag will be about ¥2000, which is a lot cheaper than the difference between the green car and the ordinary car.

The green car isn't worth it unless you're traveling during a really busy period. Unlike first class in some other countries, you don't get any food or beverages included in the price of your ticket, just a slightly larger seat.

I have been traveling to Japan for nearly 40 years, and I have never climbed Mount Fuji. If you go in March, you are 50% likely to see it from Tokyo, especially if you go to the 45th floor observatory at the Metropolitan Government building near Tocho-maé station.

The beach at Kamakura is another good place to see it, as is the Shinkansen route between Tokyo and Kyoto. Sit on the north (right) side of the train, and if it's visible, you will see it somewhere between Atami and Shizuoka.

If you go during the summer, you are highly unlikely to see it at all, even, as once happened to me, you are only 10 miles away. December through February are the best times to catch sight of the mountain.