Does a cultist's on death effect trigger during it's own turn?
It's unusual, but you're correct. The rules on page 10 state


  A monster will perform each of these abilities in the order listed (if possible) and then end its turn.


So if a Cultist has a wound and 1 HP, it will die before taking any of the actions, and thus setting up for 'on death' doesn't happen.

This is similar to how a figure with retaliate doesn't retaliate if it's killed or exhausted by an attack.

Also, remember that abilities such as disarm and stun also prevent the 'on death' ability from taking place if those status effects were in place when the cultist activated.