The best airport to access Matsushima easily is Sendai International Airport (SDJ), in Miyagi Prefecture, linked with Tokyo, Osaka, Nagoya and other national and asian cities. After the landing at the airport, take the Sendai Airport Line to reach Sendai Station in about 30 minutes. The route from Sendai Station to Matsushima is explained below.
Sendai is well served by the Tōhoku Shinkansen bullet trains, that link it to Aomori in the north and Tōkyō in the south, each in about 90 minutes. The fare for a one-way ticket is about 10.000 yen.
Once you got to Sendai Station, take a Senseki Line‘s train heading to Matsushima Kaigan Station. The platform is underground, so you cannot mistake.
All this route is covered by the Japan Rail Pass and the East Japan Rail Pass.
※ Note: Matsushima Station is also a station on the Tohoku Main Line, however it is quite far from the seaside, so be sure to get off at Matsushima Kaigan Station, on the Senseki Line (light blue, or manga painted trains).
・By highway bus
If you want to save up some money, you can get to Sendai by express bus coming from every direction: it takes about 5 hours from Tōkyō, and about 12 from Kansai area. From Tokyo also morning services are available.