Shikoku: Ehime Prefectural Museum of Art
Matsuyama has many museums, mostly located in the vicinity of its castle. The Ehime Prefectural Museum of Art has a pretty good collection of art, from both regional and European artists. It's no Louvre, but for only 300 yen, it was a good start to my day in Matsuyama before turning to the more-historical attractions.
The art museum is located about five minutes on foot from the Manamihoribata tram stop, served by all five tram lines, just southwest of the castle. While I started my morning here, it might be better to begin at Bansuiso and Saka no Ue no Kumo Museum if those are part of your itinerary. The trail up to Matsuyama's castle from Ninomaru begins near the art museum. Note that there is another path up the eastern side of the castle, so you could visit the art museum after the castle.
Museum hours are 9:40-18:00, closed on Mondays (or the day after if Monday's a holiday) and around the end of the year.