The closest apartments would be The Greens or The Marina.
At that price, I think Marina would be your best option.

There a lot of apartments to choose from.
The JBR (Jumeriah Beach Residences) have good facilities, and more shops are opening daily, though you may not get 3 bed. (your budget should certainly get you a 2 bed though)


From the Marina, you are close to both Mall of the Emirates, and Ibn Battuta (2 big malls)

Also had a thought...maybe also try Discovery Gardens (behind Ibn Battuta mall)