Bruno is the Fighting-type expert, although his first Pokémon isn’t Fighting. The rest are, though! As such, Flying, Psychic and Fairy-types will be key for winning this battle without a sweat.
First into the arena is Onix, a Rock and Ground-type. You should know it’s weak to Ground, Ice, Fighting and Steel, and doubly weak to Grass and Water. It can use Earthquake for Ground-type STAB, Iron Tail to hit Ice-types, and Stealth Rock to damage switch-ins. Try to one-shot it with a Grass or Water-type before it uses Stealth Rock.
After that, he may send in Hitmonchan. This pure Fighting-type can use Fire, Thunder and Ice Punch. Flying-types will need to be careful here because of Thunder and Ice Punch. Meanwhile, Psychic and Fairy-types have little to worry about.
Machamp is Bruno’s strongest Pokémon. This fully evolved Machop can use Superpower for a powerful STAB blow, but it’ll lower Machamp’s Attack and Defense. Otherwise, it can use Rock Slide to bash Flying-types, and Earthquake for other purposes. Stick with Psychic and Fairy.
Poliwrath is a fully evolved Poliwag that’s now a Fighting-type. It’s weak to Grass, Electric, Flying, Psychic and Fairy. For attacks, it has Superpower and Waterfall for STAB, and Body Slam to try and paralyze. A Grass/Poison or Grass/Psychic would be great here.
Finally, Hitmonlee is another pure Fighting-type. This high-kicking hero can use Brick Break for STAB and to break barriers, Feint to try and go first, and Rock Slide to hurt Flying-types. Like with Hitmonchan and Machamp, Psychic and Fairy-types should have few problems here.