You’ll pick up this quest in Chapter 3, after clearing out Scrap Boulevard with Tifa you’ll need to complete the quest Nuisance in the Factory to unlock this quest. Once you’ve got this quest unlocked, head east to the Pillar Plaza area and approach the nice guard/mean guard duo guarding the gate to the southwest. Here you’ll find a confrontation between the two aforementioned guards and a woman named Gwen. She’ll complain about some “Cerulean Drake” in the Talagaar Factory, and mention that you need a “watch security key”, which can be found in some crate in said factory.
Enter the factory and make your way to the southwestern corner, where you found the Fire Materia during the quest Nuisance in the Factory. You’ll likely have to defeat several groups of Gorgers on your way there, complete with their wall-climbing, poison spitting habits, but they’re mostly just nuisances at this point. Smash some Shinra Boxes in the outdoor area with the vending machine and rest bench and once you have the key make your way to the building on the eastern edge of the map - the one where you fought two Lesser Drakes during Nuisance in the Factory. Use the key on the previously-sealed door to the east to open it up, make sure you’re prepared - Wind Materia is useful, as may be expected.
When you’re ready, continue east under a fence to reach a walled-off exterior area, which will serve as your arena for the fight. The Cerulean Drake will rarely come down to attack, instead preferring to fly about and use its Frigid Breath attack, which functions much like a Blizzard spell. Other than that, you’ll need to be wary of its Icy Aura ability, which will damage foes that attack it at close range while active. Aside from charging your ATB gauge, getting close shouldn’t be a priority, nor is it easy to do consistently thanks to its aerial preference. Every time you charge an ATB segment, hit it with Aero - preferably when it’s performing some attack of its own so it’ll remain stationary long enough for you to hit. After a few hits with Aero it should be Staggered, finally allowing you to approach and finish it off with melee attacks.
After the fight you’ll be prompted to return to Gwen, and unless you want to smash some more Shinra Boxes in the arena where you fought the Cerulean Drake, there’s no good reason to hang around. Anywho, talk to Gwen and after being confrontational she’ll give you your reward, a Star Bracelet.
Note: If this was your last quest, Tifa will chat with you afterwards and mention returning to the apartments. If you look at your map you should see a new marker there, labeled “Alone at Last”. These markers indicate bonus scenes, which you can anjoy by heading over to the indicated spot.