The island of Utö is in the Stockholm Archipelago South East of Stockholm. We suggest you fly to Stockholm.
From Stockholm Arlanda Airport you can take the Arlanda Express train to Stockholm City Centre.
From the Stockholm Central Station you can take a subway, bus and ferry to Utö. Use the travel planner in this link and put in Stockholm C as your departing station and Utö Gruvbryggan as your arrival station.
If you arrive by car there is a big parking lot at Årsta Brygga where the ferry to Utö, Gruvbryggan leaves from. Use the travel planner above to check the ferry times from Årsta Brygga.
The latest return ferry is at 16:45 from Utö. It leaves so that it matches the public bus on the other end that takes you to the train station.