The only way for a tourist to get to Hrísey is by ferry. The ferry "Sævar" departs from the village of Árskógssandur on the mainland regularly. Travel time is around 15 minutes.
As of July 2008, departure times from Árskógssandur were 9:30, 11:30, and 13:30 -- then hourly until 19:30 -- and every two hours thereafter until 23:30. Departures from Hrísey were 9:00, 11:00, 13:00 -- then hourly until 19:00 -- and every two hours thereafter until 23:00. After 15 August, ferries merely depart every two hours all day. Round trip on the ferry costs Kr 700 per person.
If you're driving, there is ample parking at the ferry dock (the ferry is for foot passengers only). By bus, it's a bit trickier, but buses do run from Akureyri and Dalvik to coincide with the ferry. In June 2007, the 12:35 departure from Akureyri got you to the dock in time for the 13:30 ferry departure. The 17:00 ferry connected with the return bus to Akureyri. Be sure to double-check the schedules at the tourist office before attempting a day trip. Also, be sure to call ahead (tel 461-1106) to let the bus company know you want to be picked up at the Árskógssandur ferry dock. Bus fare from Akureyri to Árskógssandur is Kr 900 one-way, Kr 1620 round-trip.
This has to rate as one of the most interesting taxis I've ever seen. I think it can be hired out either for a point-to-point journey or for a tour. There appear to be plenty of seats in the back. Inquire at the tourist office if you're interested.