Dubai has a metro system that basically runs from the airport along the coast past all the mayor attractions and sights that Dubai has to offer. It does not pass the old city centre, but Sheik Zayed Road and Jumeirah are well connected.
(The Metro in Dubai)
A single ticket costs between 2 to 8,5 AED. The trains run every 3 to 5 minutes. For double the price you can use the “Gold” first class carriage which is fun to try one time.