Well, I have done this several times in my Emerald game, and those tickets are needed for several purposes:
First, activate Mystery Gift: you need to go to any Pokemart and fill in the questionnaire on the counter. Just type the following: LINK TOGETHER WITH ALL and the man behind the counter will unlock mystery gift for you. Next, save and restart your game and you should find the Mystery Gift option in your title screen. You can click on Mystery Gift and you can download special items and Pokemon, but the Mystery Gift cannot be gotten unless there is an event.
Please do note it isn't possible anymore to trigger these events. The following items could only be obtained through download via Mystery Gift when the corresponging events took place.
Mystic Ticket - This ticket was used to obtain Lugia/ Ho-Oh from Navel Rock.
The Old Sea Map - would take you to Faraway Island to catch Mew.
The Aurora Ticket - was used to obtain Deoxys from Birth Island.
The Eon ticket - was used (only through download from Mystery Gift too) to travel to Southern Island to catch either Latios (Sapphire) or Latias (Ruby). In Emerald it depends which colour you choose when talking to your mom after defeating the Elite Four. If you pick Red, you can catch Latias, if you pick Blue, you can catch Latios.
So there you go, unfortunately you can't possibly obtain any of these Items and Pokemon anymore in the Gen.3 games because the Events took place years ago.