'Yabusame' is traditional samurai mounted archery. Although the samurai may be most famous as swordsmen, they actually got their start during the Heian Period ( AD 794-1185) a fast-moving horseback archers policing the Japanese backwoods. The modern 'Yabusame' ritual traces its roots to the first Shogun, Minamoto Yoritomo, who organized it as a means of keeping his samurai in top fighting form. Over time, battle tactics changed and horseback archery lost much of its military value and instead gained ritual importance, seen as a way to appease the Shinto gods and pray for good harvests. A straight, 255-meter track will be laid out for the rider, with 3 evenly spaced targets placed on the left side. The rider will gallop his horse along the track while attempting to hit all 3 of the targets. Even for experienced riders it is a difficult feat, and those who do manage to hit all 3 targets will be awarded a special white garment in honor of their success.
The Asakusa Yabusame Festival is held each April in Sumida Park. To access the festival, take either the Ginza Line or the Toei Asakusa Line to their respective Asakusa Stations, both will drop you off just south of the park. Once there, you can either pay 3000 yen for reserved section seats up close, or grab a free spot farther back. It is recommended you show up early for either. Attendees in the Reserved section will be given a complimentary bottle of green tea and a little bag of souvenirs. If you do buy a Reserved Section seat, be aware of two things: 1) The section is reserved, but individual seats are not- so showing up early to grab a good spot is still smart, and 2) be sure not to lose your armband ticket, they don't issue replacements. Don't stand up in your seat so as to not block the view of people behind you. There will likely be a significant number of foreign visitors, and the festival staff plan accordingly- there are a large number of English-speaking staffers on-hand who be carrying a sign to identify themselves. There will even be an English speaking announcer who will translate the remarks of the Japanese MC.
The main event lasts for an hour and a half, with nearly the first 45 minutes being taken up by various ceremonies, such as having the riders parade down the track and the ritual prayers. After that, you'll be treated to more than 20 riders racing down the track (not all at once, obviously) as they continue an 800 year old warrior tradition. All riders and staff attendants are in 12-century costume to further increase the festive atmosphere (even down to wearing tachi style swords, a predecessor to the katana, history nerds like me gush over little details like that). During a lull in the action, perhaps you can take a look behind you to see the Tokyo Skytree across the river and contemplate the juxtaposition of old and new Japan.
Here's a video explaining more about Yabusame.