When I first heard about the Aquila dungeons, since my wizards were already past, I just figured I would do Atlantea when I reached the level, but when I reached level 70, I didn't get a call or something, if I need to do Mt Olympus first I can just do Waterworks a few times, cause I don't want to spend those Crowns, But I do want to try Tartarus when I reach level 90.
Pretty sure you need to complete each in order to trigger the quest for the next. You could however rent Olympus instead of buying it, to complete it and move on to Atlantea. Would save you a few bucks, but without owning Olympus you won't be able to farm the Gladiator for the Alpha and Omega Ring which is pretty much the best in the game.
I know for sure each is sequential, Olympus before Atlantea then Tartarus. But Wraithcaster has it right, the best ring and athame are dropped my the Gladiator. Each area has a special level 90 boss fight after completing the dungeon that you will want if you are a completionist and want all the badges.