The museum is open 8:30-17:00 May through October, and 9:00-16:30 the rest of the year. The museum is about 5-10 minutes from Tsuruga Castle on foot; there is also a stop on the tourist loop bus right in front of the museum.
Fukuura-jima is accessed from a bridge on the north side of Matsushima town, and the bridge toll is 200 yen (you pay when you enter from Matsushima). The entire island can easily be walked in an hour, although avid photographers and picnickers will stay longer. It's open from 8:00-17:00 (16:30 November through February).