These are the calculations for catching a shiny Lugia at a few different percentages of health. In addition, here is the catch rate calculator I used to help me. Also, these chances are rounded to the nearest hundredth.
With No Status Condition
At 100% Health: 1.57% Catch Chance
At 75% Health: 2.12% Catch Chance
At 50% Health: 2.63% Catch Chance
At 33% Health: 2.97% Catch Chance
At 25% Health: 3.14% Catch Chance
At 10% Health: 3.39% Catch Chance
At 1% Health: 3.56% Catch Chance
So overall, while it is possible to catch Lugia with a Premier Ball, it is quite difficult and may take many attempts. However, putting Lugia to sleep will increase it's catch chance to:
At 100% Health: 3.12% Catch Chance
At 75% Health: 4.22% Catch Chance
At 50% Health: 5.23% Catch Chance
At 33% Health: 5.9% Catch Chance
At 25% Health: 6.2% Catch Chance
At 10% Health: 6.74% Catch Chance
At 1% Health: 7.08% Catch Chance
With the help of sleep, the chance of capturing Lugia nearly doubles (or increases the odds by 1.99, to be exact). This allows the Pokémon to be captured more manageably. However, sleep only lasts a few turns, so you likely would need a Pokémon with Spore, such as Breloom or Parasect. If you need the catch rate of Lugia at a specific percent other than the ones listed, please let me know. I Hope I Helped!
Also, I used this source to find out that Lugia is encountered at level 50.