Instructions
Travel to Canalave City and look along the water's edge for a boat next to a house. Enter the house and talk to the sick boy and his mother. Exit the house and talk to the nearby sailor. When he offers you passage to Fullmoon Island in an effort to retrieve the Lunar Wing and save the boy, accept his offer.
Walk through Fullmoon Island's path and you'll enter a grove. When you approach Cresselia in the middle of the grove, she'll flee. Open the Marking Map Poketech app. Look for her on the app, then leave Fullmoon Island.
Open your Marker Map again after leaving the island. You'll notice Cresselia may have moved. She randomly changes location as you enter and exit different routes and cities. Psypokes recommends finding a place where two routes meet and walking back and forth between them, checking your Marking Map each time. She'll eventually show up on the route you're on.
Walk up to Cresselia once you're on the same route as her. Press "A" to initiate a battle. Cresselia may try to flee the battle. Prevent her from fleeing by equipping a Pokemon with the move Scary Face and setting it as your first Pokemon. Use Scary Face first, as it prevents any Pokemon you're battling from fleeing.
Throw high-powered Pokeballs, such as UltraBalls, at Cresselia once her HP is low. Carolyn Gudmundson's "Legendary Pokemon Guide" on GamesRadar suggests using your Master Ball, since Cresselia is so rare and difficult to catch.