Eastport requires commitment. There is no train, no bus service of any note, and no commercial airport nearby. The closest airports are Bangor International (about three hours by car) and Presque Isle (about two and a half hours). You will need a car.
From Bangor, take US-1A north to Ellsworth, then US-1 along the coast to Whiting, and finally Route 190 east across the causeway to Moose Island. The final stretch of Route 190 — the causeway crossing — signals that you have arrived somewhere genuinely apart from the rest of Maine.
From the Canadian side, the ferry from Deer Island, New Brunswick, connects to Eastport's waterfront in summer. This is a small passenger/vehicle ferry and times vary; check the East Coast Ferries schedule before relying on it.