Just do like that:
if(on_ladder):
if(ladder_up):
print("going up")
hero_sprite.play("going_up")
elif(ladder_down):
print("going_up")
hero_sprite.play("climb_down")
else:
print("idling_on_ladder")
hero_sprite.play("on_ladder")
Your codes is wrong because first you have started the on_ladder animation then started going_up animation and in the next frame of the process function, it playing the on_ladder animation again so that going_up animation stopped after a single frame. This topic is similar like this. It is a common mistake