Mystery Gift in Pokémon FireRed (GBA)? Impossible - Here's Why

Impossible. Mystery Gift does not exist in FireRed/LeafGreen (Generation 3).

Generations With Mystery Gift

• Gen 2 (Gold/Silver/Crystal): Infrared (GB Color) / Mobile Adapter GB (Japan only).
• Gen 4 (Diamond/Pearl/Platinum/HG/SS): Nintendo Wi-Fi Connection (defunct) / Local Wireless.
• Gen 5+: Internet / Local / Serial Codes.

FireRed/LeafGreen Connectivity

• Link Cable (Wired): Trade/Battle with another GBA + FireRed/LeafGreen/Ruby/Sapphire/Emerald/Colosseum/XD.
• Wireless Adapter (GBA): Same as cable, but wireless (short range).
• No Internet / No Mystery Gift.

Getting Rare Pokémon in FireRed Today

• Trade: From Ruby/Sapphire/Emerald (Hoenn starters, legendaries, Johto starters post-Elite Four).
• Historical Events (Unobtainable Now): Aurora Ticket (Deoxys), Mystic Ticket (Lugia/Ho-Oh), Old Sea Map (Mew) — distributed at physical events 2004–2006.
• Cheats / Save Editing: Only way today. Flags for Mew, Deoxys, Lugia, Ho-Oh via Action Replay or VBA save edit.

Focus on the rich post-game content: Sevii Islands, Cerulean Cave, Trainer Tower. That's the intended experience.