OK so...
MISSION 1: Through the Nintendo Wi-Fi Connection, there are 6 other-colored Pokémon available in all for you to get!
MISSION 2: With your Dragon-type Pokémon, battle intensely with Pokémon to make it grow!
...by the looks of it, it IS a simple Wi-Fi event (as in, not a local-only event, but one that can be done over the internet). I thus feel it's safe to assume that the given dates are when that particular Pokémon is made available, meaning you have a chance to get all six of them!
Lemme find out more about this though... Maybe you can only get one in all, and you just have to wait for the date to get the one you want....
...
EDIT: OK, I'm thinking it's not so clear cut. Right above the dates there is a section called "Precautions", and there is a word colored red which means "randomly". Secondly, next to each number is the character "회", which means "time" (as in, first time, second time, third time, etc).
Therefore, I suspect that you have six chances to get a Pokémon, and each chance happens in those time periods (once a week). And with each chance, you get a RANDOM Pokémon from the lot. Basically meaning, you may as well end up with six Horseas by the end of this.
I'm working on the actual translation, but this is what it looks like is going on. Oh well, c'est la vie!