This is one of three main quests that will appear in your journal after helping the mysterious beggar, which means completing the "Mysterious Man" main quest.
Explore the town. You might find something interesting.
Heart Beyond Borders
Quest stages of Heart Beyond Borders
1. Speak with the child.
As usual, the location you should head to is marked on your map and is located next to the house with the bell tower. There, you will find a girl in a green cloak named Rulupee who, when spoken to, will ask for your help getting her cat down from the roof.
2. Return the cat to its owner
Climb onto the roof and follow the black cat. It will stop after a moment, so you shouldn't have any trouble catching it.
Once you catch the cat, your character will automatically get down and hand the cat to the child. She will thank you and invite you to her town before walking away, without mentioning where it is located.
You will notice she dropped something on the ground, so pick it up - it is Shai's Pendant. This will complete the quest.
You rescued the cat stuck on a rooftop and returned him to his owner. The thankful child invited you to visit their town before disappearing, accidentally dropping a pendant on the ground in their haste.