How do you get out of the zone where you meet bucky and rival

Have you beaten the elite four AND attained the national dex? Fairly certain you need to have done both before you can go any further. Need I remind you that the national dex is obtained by seeing each and every Pokemon in the sinnoh dex, including dialga and palkia and #114 which I'm guessing you are missing.

Ah yeah, I remember having the same exact problem for a while myself. Eventually, I figured out what it was that I needed to do. In order to go to Battle Island, you need to defeat the Elite Four. But in order for the people in your way to move so that you can go to other areas on the island, you need to catch all 150 Pokemon in the Sinnoh Region, and obtain the National Pokedex from Professor Rowan. After you get the National Pokedex, the people will have moved.

Hope this helps.

You first have to beat the Elite Four. Then, you have to obtain the National Pokedex. To get the National Pokedex, you have to see (not catch) every Pokemon in Sinnoh. When done so, go to Prof. Rowan, and you will see Prof. Oak there. Talk to him, and he will give you the National Dex. When you've gotten the National Pokdex, and beaten the Elite Four, you will be able to access the other areas past those people.

If you have an Action Replay, there is a cheat that comes with it that completes your Pokedex 100 percent.