Just beyond Kofukuji, in Nara Park, the Nara National Museum holds Buddhist statues, scrolls, paintings, and ceremonial objects from all over Japan and abroad.
The original building, built over 100 years ago, is still used, though you couldn't tell from the outside. Once inside, though, you are treated to a great collection of interesting Buddhist works spread over two buildings. The entrances are covered to protect the works inside, removing sunlight and opportunities for dust to enter the building and damage the artifacts. You can enter this building from the opposite side, or you can enter via an underground passage that leads to the other wing.
Many artifacts are posted on the museum's English website, but for those into Asian, religious, or Buddhist art, I highly recommend this spot. That's saying a lot, because I am not very interested in historical Asian or religious art. (I live in Japan. Go figure.)
The museum is open 9:30-17:00 (sometimes later), closed on Mondays. Admission is 520 yen for both wings. Simply walk straight from Kintetsu Nara Station toward the park; getting around Nara is pretty easy.