Bukimi Rumor Side Quest - Persona 2: Innocent Sin Guides
The Bukimi Side Quest begins by speaking with Effeminate male student on the 3F of Kasugayama High. You can find this guy standing near the stairs leading down on the 3F. During your first visit to Kasugayama High this quest will not be available to you, it only becomes available once the school faire/event starts. This doesn't happen until after you complete the Air Raid Shelter dungeon and return to Kasugayama High.
What this means is you have an extremely small window of opportunity to do this quest; this is extremely common when it comes to rumors for this game and something you'll have to get used to. Typically when you learn a rumor inside of a dungeon like this you're expected to drop everything you are doing and go spread + complete the rumor.
Once you finish the Kasugayama High dungeon you won't be able to complete this rumor anymore so make sure if you plan on doing it - you do it now. All you need to do for this quest is run around the school and get into random encounters until one of them is Bukimi. Once you defeat Bukimi give the item she drops, Bukimi's Nametag, to the same kid who started this quest (pictured above).