Kudan Side Quest - Persona 2: Innocent Sin Guides
The Kudan Rumor can be started once Sumaru City turns into Xibalba and takes off into the sky. During the main story this happens immediately after you get Jun to join your party, it's part of Chapter 4 in my Persona 2 Walkthrough. Once these prerequisites have been met you'll want to visit Sumaru TV in Aoba and speak with the Pale man standing next to the desk (pictured above).
Now that we've heard the Kudan Rumor it's time to spread it at the Kuzunoha Detective Agency, you should know the drill by now. Once the rumor is spread you'll want to head over to the Abandoned Factory and run around inside of Room #5 until you encounter Kudan the enemy. In my experience you can only encounter Kudan inside of Room #5.
Kudan has a small chance of using the Prophecy Skill which will instantly kill him but it'll reset your Persona's level back to 1! If you don't feel like grinding your Persona's again then make sure he doesn't get the opportunity to use this skill.
Defeat Kudan and you'll get Kudan's Head as the item from battle. Return Kudan's head to the man inside of Sumaru TV and you'll be rewarded with... nothing. Your reward is the friendship(s) and the experiences you had along the way.
Note: Though, the final dialogue that this guy says is actually quite insightful, "Maybe the human mouth is scarier than anything else in this world." I'll accept this nugget of wisdom as my reward for this quest, because it's quite literally true, our words can be scarier than anything else in this world.