The museum is open from 0900-1700 daily, with tickets costing 600 yen for adults, 500 for senior citizens, and 300 for children. The exhibits are in both Japanese and English, and cover a wide range of subjects. You can tour the ship itself and see the restored guns, bridge, telegraph room, admiral's cabin, among other areas. The lower decks also include a lot of information on Japanese history from the arrival of Commodore Perry in 1853 to the Russo-Japanese War in 1905. The story of the Mikasa is also the story of Japan's rapid modernization in the latter half of the 19th century- to put it in context, Admiral Togo who commanded the Japanese fleet at Tsushima was a sword-wielding samurai as young man and had to go from a medieval to a modern understanding of war in a few decades. Now, there is one area of controversy in the museum's information, as it has a clear pro-Japanese slant in justifying Japan's actions up to and during the Russo-Japanese War. East Asian history is always a hot button issue, but for the sake of this article let's simply say that whether Japan was resisting Western imperialism or whether Japan and Russia were fighting over who was going to build their own empire in Korea and China depends on who is telling the story. Regardless, the ship is a fascinating visit for any history buff, and with the dock for the Sarushima Island ferry directly adjacent to the Mikasa, it's easy to see two of Yokosuka's major attractions back-to-back.
You can find more information about the Mikasa and the Battle of Tsushima here.