First off, obtain a Tenshodo Invite. This can be purchased from the Auction House under Misc. and resold once you gain entry.
Next, go to Lower Jeuno and into the door called "Neptune's Spire." There should be an elvaan man with glasses named Faursel standing in the middle of the room.
Once you speak to him, he will ask you a series of questions. You must select the hidden answer (below all the other choices) each time.
Once you answer all his questions correctly, you will be able to choose which 'path' to take to get your Boarding Pass. Low levels are recommended to take the Beginner Path, but higher leveled players may find it faster to farm the items needed for the Intermediate and Advanced paths. Whatever path you choose, the result is the same. Alternatively, if you are rich you can choose to pay 500,000 gil to get your pass (this will warp you to wajaom woodlands).
Once this step is done, zone and wait a game day. Speak to Faursel again to obtain your pass. You may now travel to Aht Uhrgan via the ship in Mhaura.