Taking hit from Plague Zombie
You'll take damage and then trigger the "if undefeated" effect

This is covered under the Encountering A Card section on page 10 of the rulebook where it outlines the specific steps to resolve an encounter, and under the Attempting A Check section on pages 10-12 which details out one of those steps with several sub-steps. 

The steps to resolve an encounter are:  


  Evade the Card (Optional)
  Apply Any Effects That Happen Before The Encounter, If Needed
  Attempt the Check
  Attempt the Next Check, If Needed
  Apply Any Effects That Happen After The Encounter, If Needed
  Resolve the Encounter  


When you get to the Attempt the Check step, you follow these sub-steps:  


  Determine Which Die You're Using
  Determine the Difficulty
  Play Cards and Use Powers That Affect the Check (Optional)
  Assemble Your Dice
  Attempt the Roll
  Take Damage, If Necessary  


So you'll take the damage from failing to defeat the Plague Zombie first during the Attempt the Check step's Take Damage, If Necessary sub-step. Then, during the Resolve the Encounter step, you have not "succeeded at all of the checks required to defeat a bane", so it is undefeated, triggering the special effect on the card.