In Dungeon Fighter, what is the priority between death and activating an ability?
Quick disclaimer, I've not played the game or even heard of it until I read the question.   However, I've watched a tutorial video and found the rules online here  to try and find the answer for you.

Look at the rules on page 9 for 'No Damage'  it says :-


  No damage: If activated, the hero takes no damage if he misses the target board.


This might infer that you get to activate this ability before taking damage.  However unhelpfully back on page 7 the rules say this :-


  Hint: The damage dealt while becoming defeated by a
  monster is not avoidable. No special powers can prevent that
  damage. 


It would be really helpful to have that 'hint' on a page with all the abilities but it does seem to make it explicit that if a monster defeats you then you do not get to activate a special power, including those that are activated when a character misses.