My life would suck without you
To get to Lugia, head right from the starting point of Island B's entrance. Jump across the first ledge and keep to the upper path; there are two adjacent ledges coming up and you want to jump across the upper one, finding a ladder located on regular ground to the north. If you find a ladder near an Ultra Ball on a higher platform, you jumped across the wrong ledge and will have to start over. Once you ascend the right ladder, follow the raised path and you'll see the next ladder you need to take on the southwestern end.
You'll emerge in an area where Flash is not required, near a Max Revive; walk south and you'll find a monk blocking the path. Speak to him and you'll produce the Silver Wing, which demonstrates that you're a pure-hearted Trainer and ready to encounter Lugia. The monk will move out of your way.
Pop into the cave nearby for a free Rare Candy, and then continue all the way down into this cavern. When you reach the entrance to Lugia's lair at the foot of the path, your Pokemon will return to its Pokeball as you head inside. The Kimono Girls will be waiting for you and, using the Tidal Bell, they will perform a dance that summons Lugia from the water below. Lugia emerges from a waterfall, flapping its wings and roaring ferociously. After the cutscene, save your progress, Surf out to Lugia and speak to it to begin the battle. This is your only chance to capture SoulSilver's mascot, so hopefully you brought some Ultra Balls and a Pokemon that knows a sleep-inducing move.
That's from Psypoke, hope it helps.