The S.S. Ticket is necessary to board the S.S. Aqua, a ship that sails from Olivine City in Johto to Vermilion City in Kanto. You’ll get the S.S. Ticket as a gift from Professor Elm at his lab in New Bark Town after you’ve defeated the Elite IV and Champion Lance.
In other words, the S.S. Ticket is really your ticket to the second half of the game.
Prerequisites: Beating the Elite IV and Champion Lance
The Elite IV and Champion Lance can be fought at the Indigo Plateau after you collect all eight Johto Gym Badges.
To do this you’ll have to use Surf to go east of New Bark town onto Route 27. Then you’ll need to use Waterfall to cross Tohjo Falls.
Continuing in this direction will take you through Routes 27 and 26, the Pokémon League Reception Gate, and finally through Victory Road before reaching the Indigo Plateau.
But basically, after you’ve beaten the game you’ll be able to get the S.S. Ticket from Professor Elm.
Getting the S.S. Ticket
After defeating the Elite IV and Champion Lance, you’ll be returned to New Bark Town to embark on your post-game journey.
Prof. Elm will give you a call on your PokéGear telephone and invite you to his lab in the northwestern part of town.
So pay him a visit in his lab – you’ll find Prof. Elm in the back of the lab as usual.
Talk to him and he’ll give you the S.S. Ticket.
Using the S.S. Ticket to Visit Kanto
To use your ticket to board the S.S. Aqua, you’ll need to make your way to Olivine Harbor.
You’ll find the entrance to the harbor directly southeast of the Pokémon Center.
Follow this path all the way down until you reach a pier.
A sailor from Fast Ship S.S. Aqua will ask if you’ll be boarding today. Say “Yes” and show him the S.S. Ticket and you’re off to Kanto’s Vermilion City.