Samurai are one of the most famous cultural exports from Japan. For me, I have been fascinated with samurai culture ever since reading Shogun by James Clavell nearly twenty years ago. (It remains my favorite book to this day.)
The samurai period ended with the fall of the Tokugawa Shogunate in 1867, but the concept of honorable single-minded warfare and a strict code of living remains fascinating even today. Cartoons, action figures, comic books, and other media portray the samurai lifestyle in fantastical ways, but a visitor to Tokyo can experience the real thing for himself.
In the Shinjuku neighborhood (also home to many of the popular western hotels) you’ll find the Samurai Museum. It is fairly small, but made up for by free 45 minute guided tours (with paid admission, of course). Visitors will be led through both original and replica armor, swords, guns, and other samurai artifacts, hearing stories from the Edo period (that of the aforementioned Tokugawa dominance). After, you will have the chance to purchase a replica sword to be shipped home to you, starting from around $100.
If after seeing the museum you want to get a bit more personal, head to HiSUi Tokyo for classes on traditional Japanese art forms. Samurai were expected to be masters in all areas of Japanese tradition, and at HiSUi, one can learn the art of kimono, calligraphy, tea, and the katana. Each class is about 45 minutes long, and consists of both demonstrations and participation. So you can make and serve tea in a traditional tea ceremony and then learn to wield a katana (they are incredibly sharp so adults only).
While you may never get to be a real samurai, these experiences will enable you to feel like one for a couple of days!