To the Bat Cave!: wipe out a nest of vampires.
You enter this zone from the south into the crevasse between two ledges. There are bats on the upper levels that may shoot fireballs down at you. Follow the tunnel at the northern end of the crevasse to reach the western ledge, and cross over the bridge to the other ledge to reach the door to Zone 2.
There are no vampires in this zone, only four bats.
You enter this zone from Zone 1 on a ledge overlooking the wide crevasse in front of the main door to the old castle. There are bats on the ledge with you. To get to the lower level, you can jump down, or go through Zone 3. You'll have to go through the castle to get back up.
The Vampire Patriarch is in the cave at the top of the map, rather than in the castle, but you would normally go through the castle (Zone 3) to reach him. He's standing next to a chest containing enchanted arrows. Take these back to the castle as part of the quest.
There are seven bats, five vampires, and the vampire patriarch in this zone.
You enter this zone through the door at the left from Zone 2. All the enemies (11) in this zone are vampires, each with a 50% chance of appearing.
You can exit to Zone 2 at two locations. The central one takes you to the lower level, where you'll find both bats and vampires, probably fighting each other. The right end door takes you out to a ledge where two vampire archers are waiting.