Hi Ko,
In Rahat market is a bike shop, selling western brand bikes. You can find them on the 3rd floor, shop nr. 336 or 338.
On Dina market you can find cheaper bikes, which usually come from Chinese factories.
Good luck.
OEPS sorry, just realised now you're looking for a motorbike!!! Well, that will be difficult. No shops selling them here. There are only a handful of motorbikes here in Atyrau, so you need luck to find someone who wants to sell it to you.