According to IGN, the only official way to get the Mystic Ticket is to attend a Nintendo special event where you can download the ticket or receive a code for it. After the item is downloaded, the Mystic Ticket appears in your inventory.
There are no more "Pokémon Emerald" events scheduled in the United States as of 2014. However, you can still get the Mystic Ticket by entering an unused download code. You can also achieve the same effect as the Mystic Ticket (access to the Southern Island and the rare Pokémon there) using a Gameshark or other cheat code, but you cannot get the Mystic Ticket itself in this way.