While most anime is focused on modern-day Japanese life, or life in fantastical worlds, there's actually a lot of anime with roots in traditional Japanese culture. These shows not only have great stories with compelling characters, but they're also a wonderful way to learn about real, traditional Japanese culture (without having to crack open a textbook).
Do you want to learn about rakugo, a form of Japanese storytelling? Try Showa Genroku Rakugo Shinju. How about shogi, a form of Japanese chess? Try March Comes in Like a Lion. There's anime about traditional Japanese culture of every kind, so choose the one that best suits you.
Rurouni Kenshin is far from the only anime that focuses on samurai, but it's one of the most well-known. Samurai were members of the warrior caste in medieval and early-modern Japan. They generally belonged to clans, and were expected to serve a daimyō, or a feudal lord.
They followed Confucian teachings, and their behavior was often used as a model for other castes and classes.