Instructions
Travel to Olivine City and choose a Pokemon to surf in the water; head south in the water until you reach the top-right island.
Use Whirlpool to enter that particular island. Head onto land and enter the cave.
Travel to the upper-right ladder in the first room of the cave. In the next room, proceed along the walkway to the next ladder. Enter the water in the next room and travel down the waterfall. At the bottom, there is another cave entrance, which is where Lugia can be found.