With the Aurora ticket it is possible to travel to the island of origin and challenge deoxys there. In PokéOne this is a consumable item as soon as you travel to Birth Island it is consumed. You can get it daily in the Trainer Challenge (here is the guide). Or you can buy it for 50.000 Pokédollars.
A Bicycle is a Key Item. The Bicycle allows for faster traveling than walking. It also allows the player to pass the Cycling Roads on Route 16 to 18 in Kanto.
In Kanto you need a bicycle voucher from Vermillion which you can exchange for a bike in the bike store in Cerulean (Guide).
In Johto you can get the bike easily without any requirements. Just go to the bike store in Goldenrod and you will get one for free. Also for this we have a small picture guide, so you can find it easier (Guide).