On this page of the Tunic guide, you will learn the lantern's location and how to find it. This item will illuminate the rooms that you will go through.
Go to the first shrine (checkpoint) located in Overworld. Then go down the stairs to the left of it. Next, turn right between the trees and the wall near the small waterfall. This is how you discover the secret passage leading to the Flooded Well.
But before you get there, you have to make a quite long way. Stick to the narrow path next to the wall until you come across a group of Rudelings. On your way, you will find a passage to the Dark Tomb, but ignore it for now. Deal with them, then go through the wooden footbridge to the stone area on the water. In front of you, you will see the well mentioned before, into which you have to jump. Watch out for Autobolt, which will be shooting at you.
At the bottom of the well, you will find the Under the Well location, which you will have to explore if you want to get to the lantern. It is a small maze of paths filled mainly with Slorms and Rudelings. Explore the rooms, killing enemies one by one, and try to avoid the water, in which you can easily die due to the tentacles lurking in it.
Keep going through the Under the Well until you find a room with a ladder and skulls on the ground. Deal with the Phrends hidden there, then enter the tight dark tunnel.
However, don't worry and keep going until you find the aqueduct outside. It will lead you to a cube emitting purple light, next to which is the lantern.