Hi! Sorry. >.< I realized this was asked hours ago, and I don't expect anything from answering this question, but I though I'd give you a full answer that would hopefully answer any other questions about the Bug Catching contest- for both this question and future questions.
Join the Bug-Catching Contest in the National Park
The catching contest is held almost every other day: Tuesday, Thursday, and Saturday
The National Park has a Bug-Catching Contest every Tuesday, Thursday, and Saturday. You can take part in the catching contest once on each of those days, competing for rare items and Bug-type Pokémon.
1st Place: Sun Stone
2nd Place: Everstone
3rd Place: Sitrus Berry
Consolation Prize: Shed Shell
Rule #1: You can only use the Pokémon at the head of your party
You can only use your lead Pokémon for the catching contest. The rest of your team is left at the reception deck. You'll only encounter Bug-type Pokémon, so bring a Pokémon that's good at catching them.
Use moves like False Swipe to whittle down HP
It's a good idea to bring a Pokémon with moves that inflict status conditions or drop a target's HP into the red, like False Swipe
Rule #2: You only have 20 minutes to catch Pokémon
In the Bug-catching Contest, you get 20 Sport Balls. You have a 20-minute time limit and a single goal: catch the best Bug-type Pokémon you can! Whenever you catch one, you have the option to swap it for the one you've already caught.
You can't use any items, only the new Sport Balls.
Swap Pokémon to keep the strongest-looking one.
Rule #3: Your final Pokémon determines the winner
The catching contest is over when time is up, or you run out of Sport Balls. Final placements are determined by comparing the Pokémon that each Trainer has caught. To win, catch the Pokémon with the highest level and point value.
Now to actually answer the question. :P
The Point-Values per Pokémon:
- Beedrill (Avg.)
- Butterfree (Avg.)
- Caterpie (Low)
- Kakuna (Low)
- Metapod (Low)
- Paras (Avg)
- Pinsir (High)
- Scyther (High)
- Venonat (Avg.)
- Weedle (Low)
That was alphabetical. Here is by category:
Good Contenders for each Category:
- High-point-value Pokémon
- High-lever Pokémon
- Shiny Pokemon
Meaning for all of this:
You have 20 minutes and 20 Sport Balls to catch either Scyther/Pinsir at a somewhat high level (shiny is a nice factor), and you should be getting good results each time. Or, if you want other prizes, you can mix and match as you feel
Entire answer plagiarized from The Official Pokémon Johto Guide and Johto Pokédex (HeartGold SoulSilver).