You will be staying in a good location. The Shinagawa Prince hotel is right across the street from Shinagawa station. The Yamanote train line stops there, and on the Yamanote train you can visit most of the "must see" places in Tokyo such as Ueno, Akihabara, Ikebukuro, Shibuya, Harajuku, Shinjuku, and Tokyo Station.
Ueno is the home of the Tokyo Zoo, the National Museum, the Museum of Western Art, and the Toshogu Shrine. These places are all closed on the first Monday of each month, so plan your trip accordingly. In March and April, Ueno is the place to be for Hanami, or cherry blossom viewing, but it's a beautiful place to visit any time of year.
Akihabara is also known as "Electric City", and is the world's headquarters for the newest in gadgets. Don't expect to find any good deals unless you are a careful shopper. The best deals will be found in the alleyways in small shops rather than in the larger department stores along the main streets.
Akihabara is also the anime/manga/video game headquarters of Japan, if you are into those things, you'll find enough there to keep you busy for months. You must have coffee in a Maid cafe, look for the hot girls in maid outfits in the area around the station for directions to the nearest one.
Ikebukuro is the home of Sunshine City, which is Tokyo's tallest building. It is also the home of Namjatown and Ice Cream City. Like many other parts of the metro area, Sunshine City sports a large shopping mall with a huge selection of shops.
I'm sure Shibuya and Harajuku need no introduction. They are almost always on the top of the list of things for foreigners to see when in Japan. Be sure to visit on Sunday if at all possible.
At Shinagawa there is an aquarium adjacent to your hotel. It is worth seeing, so check it out if you have the time.
At Tokyo Station you'll be within walking distance of the Imperial Palace and Ginza. If you are going to Ginza, be sure to bring a no-limit credit card, fashion items there tend to cost about 40% more than they would an any other major city. Take a look at the Kabuki theater on Harumi-dori street. If you head a little further east you'll come to Tsukiji, the world's largest fish market. It is surrounded by small food stalls and restaurants, all are exceptionally good, regardless of appearance.
A couple stops from Tokyo Station is Shimbashi. Get off the Yamanote train and get on the Yurikamome train going to Odaiba. Be sure to bring your camera, Odaiba has some of the best views of the city, as well as of the rainbow bridge. Take a ride on the giant ferris wheel, the cost is 800 yen, and the ride takes 15 minutes or so to go around, you'll enjoy even better views from the top.
From Odaiba you can take the water bus to Asakusa. The Sensoji temple is located there, and it is one of the oldest temples in Japan. There is a huge market place surrounding the Sensoji, and several shops sell yukata and kimono. Some take credit cards, and some don't, so be sure to bring cash. There are two types of yukata, the standard type which looks like a kimono, and a short two piece version which comes with shorts known as "jinpei".
From Asakusa station you can catch the Tsukuba Express, which will take you back to Tokyo Station, from where you can once again transfer to the Yamanote line.
Bring comfortable, well broken-in shoes. You'll be doing a lot of walking. Cash is more convenient than credit cards, and safe to carry. Before you go into a nice restaurant, be sure ot look at the menu outside. Though most restaurants are reasonable, some are outrageously expensive.
I don't really recommend Roppongi as a place for tourists to visit. On one side there are numerous seedy bars and clubs which garden variety foreigners visit; Nigerians stationed outside these places will try to hustle you to go inside, or to sell you dope, or whatever. On the other hand, if you are interested in meeting Japanese women, many who like foreigners frequent. On the other side is Roppongi Hills, which is home to the wealthier expats who work in Tokyo's financial sector.
Enjoy your trip, I'm sure you'll love your time here
