Traditional Noh and Kabuki theater are, and specially the Noh theater, often incomprehensible by simply listening to it for Japanese people as well. They use very much modulated sounds, ancient characters that are only understood 100% by scholars and people mostly understand it because they are familiar with the famous storyline.
Many big theaters in larger cities offer printed translations and even live audio translation in several languages to make you understand the text spoken in real time. See this guide for availability.
Other types of traditional culture you can enjoy are traditional sports such as Sumo and Kendo fights. Since neither of them require a lot of language, they still have a large importance in Japanese culture. On top of that there are many many festivals every month showcasing local traditions in many Japanese towns that are very much worth watching. Some of them involve break-neck speed racing of heavy festival carts through cities - even in winter, half naked - and many other race/sports like activities you will only see on one day of the year in one small town.
What is interesting to know I found is that those theaters when performed for nobles used to last a whole day where people came in and left as they pleased to see special scenes that they liked most. Still today you will see people walk in and out of performances as if nothing happened. Same by the way for Sumo where a sumo match could go on long into the evening if the fighters chosen to start the fight used delays as a psychological weapon to grind down the opponent. Only the advent of modern schedules and in case of Sumo of the evening news on TV made those traditional performances shorten themselves down to a couple of hours (in case of Noh 1.5 /Kabuki 3) or a whole day until 8:00 in case of Sumo.