Just go over to the registration renewal place in Al Barsha. You'll have to bring passport and visa copy, as well as your current registration and your new insurance confirmation letter. It is pretty easy.
Be warned though, I've heard the Dubai registration center requires you to have a triangle in your car and that you tires be in really good condition. Make sure you have a triangle (borrow one - don't buy an expensive one onsite) and hope for the best with the tires.