Same basic strats apply to all alien rulers for the most part. Alien hunter weapons are very effective so try to have them upgraded as much as possible. Rulers get a turn for every time you spend an action token, so if you move they get a full turn, shooting certain weapons or using certain gear will also give them a turn. Poisoning and Burning rulers is also effective to stack some damage on them. Repeaters work on rulers so there is a small chance to instantly kill a ruler.
*** Spoiler - click to reveal ***After losing a certain amount of hp a ruler will flee and appear in another mission, but it will retain any damage received from previous encounters. By the time you reach the archon ruler you should have the Serpent suit and Rage Armor. Using their abilities is free and doesn't give the ruler a move, same for shooting[ the Boltcaster (not reloading it) , and throwing the axe, not using the axe melee. Frost Bomb will incapacitate it for 2 soldier moves, as will frostbite. Remember to use the end turn button, as putting a soldier on overwatch or even taking a shot is sometimes not worth giving the ruler free moves, as even walking and overwatching gives them a turn. The boltcaster has a decently high chance to stun when used against an alien ruler. Try to only engage the ruler with its own pod. If there are multiple enemies with the ruler try to take them out with AOE damage that hits the ruler as well. Slowly whittle away at the Rulers over multiple encounters as once they lose a certain amount of hp they will just try to flee.