Yellow taxis are almost always available to take you anywhere in town - and beyond if you like. The standard fare should be 1 JD, though they might ask for more or tell you to pay however much you like.
Jett operates 3 buses a wek from close to the Abdali Station in Amman (but not exactly in it...the Jett station is 1km up the road). It leaves Amman at 6:30 am and leaves Petra/Wadi Musa late in the afternoon around 4. The trip takes 3.5 hours. This is by far the cheapest option available. Round trip is 12 JD.