You can only renew your license in Madinat Khalifa or any other traffic department. The new plate is fixed at Duhail, irrespective of the traffic department you renewed your license. You need to wait at least 2 weeks before going to Duhail after the renewal of your registration. If you didn't claim it in 30 days, they will charge you for license plate, otherwise it's free.
As far as I know, they operate on Sat morning and full day (till 6 pm, I guess) on week days.