No, the Abra will be born with the standard or DW ability that Abra can have, not the ability of the other species of Pokemon, like Prof said.
Here's the way that you get a DW Ability, though.
Getting started... You'll need to go to the following site:
And from there, you'll need to log in with your Pokemon Trainer Club Account. You may need to create a new username for your Pokemon Global Link Account, so you can either make up a new one or use the same username as on your Pokemon Trainer Club Account, it's up to you.
You'll also be prompted to enter the Game Sync ID of either your Pokemon Black or White game. That is required in order to receive any items or new Pokemon that you earn through playing the mini-games in the Dream World. So you should get this done now, or whenever you have the chance.
After you successfully join your two accounts and register your Pokemon Black or White Game Sync ID, you'll be brought back to the Pokemon Global Link main page, and from there... You can do whatever you want! Well, or at least you have a number of things to check out! And one of them is the Pokemon Dream World!
Enjoy my little tutorial of the Pokemon Global Link!
There are lot's of videos of it on Youtube.
Hope this helps.