To catch the Legendary Pokemon Lugia in Pokemon Emerald, the player must first gain access to Navel Rock, and then take the path to the right inside the cave to get to the Pokemon and capture it. The only way to gain access to Island 8 Navel Rock is to obtain a Mystic Ticket from a Nintendo Event. Apart from Lugia, the Legendary Pokemon Ho-Oh can also be caught on Navel Rock.
The following shows how to get to Lugia in Pokemon Emerald in order to catch it.
Once the player gets on Navel Rock, he or she must enter the cave and follow the path until the player gets to the stairs.
There is a t-junction on top of the stairs. The player should take the right path.
The player will come face to face with Lugia, at the end of the path, where he or she can get the chance to capture it.