The Mystic Ticket is only available from Wonder Spot machines. These are special machines that Nintendo brought to Pokémon events and temporarily made available at certain Toys "R" Us locations and Pokémon Centers in the mid-2000s.
Players who encountered a Wonder Spot machine could connect to it using the Game Boy Advance's wireless connection and download the Mystic Ticket to their game by using the Mystery Gift feature.
Nintendo still holds periodic Pokémon events for their newer games, but wireless downloads for the older games such as FireRed may no longer be made available there. If this is the case, the only way to obtain the Mystic Ticket would be to trade with another player or hack the game code using an Action Replay or Gameshark device. It is also possible to enter cheats in this manner that allow the player to reach Navel Rock, which is the purpose of the Mystic Ticket.
Navel Rock is home to two rare Pokémon, Lugia and Ho-oh. Both of these Pokémon are at level 70 and are formidable opponents. If the player already has access to these two Pokémon through other means, then there is little point to obtaining the Mystic Ticket as Navel Rock contains nothing else of significance.