The way I read the rules you have to "call your monster" before the first monster is reveled. Like you know that a dragon went into the dungeon and someone just removed the Dragon spear you could say the vorpal axe kills dragons. Then when the dragon card is revealed he is killed just like if you had the spear.
You are thinking of the Vorpal Sword, which you do have to declare a type of monster before entering the dungeon and the sword will defeat those monsters.
The Vorpal Axe, on the other hand, states to defeat a monster AFTER you draw it, so you can choose when to use it while going through the dungeon.