Everyone who visits Japan wants to see a Sumo tournament — it is uniquely Japanese. However, this is relatively difficult. There are only a few tournaments a year and tickets sell out fast. They can also be hard to obtain as a foreigner. I was lucky to go twice, but trust me, this was really lucky and entirely due to the kindness of others. However, there are some other special Japanese traditions you can experience more easily.
If you want to stick with sports, go to a Japanese baseball game. Baseball season is spring through fall so you have a much better chance of getting tickets. And while baseball did not originate in Japan, they have taken it to a completely new level.
If in Japan in July or August, attend a local Tanabata (star) festival. Tanabata is an ancient story dating back to the 8th century that celebrates star cross lovers separated by the milky way. They can only unite on the 7th day of the 7th month. If it is raining, the water washes out the bridge, the couple will remain apart for another year. Every town has their own festival (so the dates are spread out through July and Aug) and are celebrated with decorations, food and music. It is a family event that should not be missed.
Every Japanese festival (including Tanabata and especially the Bon Odori) includes Bon dancing. You can consider the Bon dance as a folk dance that Japanese learn from the time they are young. But there are of course varying levels of proficiency. You can even take classes or sometimes join in on the dancing.
Taiko Drums are also common at festivals and uniquely Japanese. I am always so impressed at the coordination, energy, and collaboration of the musicians.
Similar to Sumo, many visitors try to time their trip for Cherry Blossom (Sakura) season. While cherry blossoms are indisputably beautiful, springtime is amazing in Japan and there is no bad time. If you miss the cherry blossoms, you will see the Plum Blossoms, Azaleas, or Wisteria.