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Free internet access at two computers is available at the Miyanoura port terminal "Marine Station Naoshima," 8:30 to 18:00, not all during its operating hours.
Each division in the port terminal has different business hours, so pay attention
There is a 24-hour Internet & Comic cafe one of the Venite chain at the Uno port on Honshu look for the Tsutaya building from the port/train station, it's on the 2nd floor but expect trouble from the staff as one has to sign up and obtain membership before use.