In order to find the mythical bird you must first satisfy 2 conditions:

1. Snag and fully purify all 48 Shadow Pokemon in story mode.
2. Beat all 100 trainers on Mt. Battle. This can be done in either Single Battle or Double Battle mode.

However, you can only use a team from story mode, not from your Ruby or Sapphire game pack. If you want to get around this rule, simply use the trade center in Phenac City to import your Pokemon into your story mode team.

After beating the 100th trainer on Mt. Battle, Ho-oh will show up and will automactically be transported to your PC.

Note: If your PC is full, you will not get Ho-oh and must retry the Mt. Battle challange.