You have to speak to the outrider NPC first in order to "join" the invasion event. Otherwise, it won't count as completed for you, and you can't open the chest at the end.

There are no in-game achievements for Imperial Invasions, only Lodestone History entries, which can sometimes take awhile to show up. The chest usually gives you some potions and deaspected crystals.

The NPC disappears when the event is over, but there are usually several "rounds" to an invasion. There's about 15mins (I think) in-between rounds.