The standard route is to fly from Yangon or Mandalay to Sittwe, the Rakhine State capital, and then take a boat up the Kaladan River to Mrauk-U — a journey of around four to five hours depending on the vessel. Boats run daily when water levels allow. There is also an overland route, but the road is rough and the journey much longer; most travellers take the river.
Note that foreign visitors to Rakhine State have at times required special permits, and access restrictions have changed. Check current entry requirements before planning your trip.