Aqaba to petra

All I can tell you its 127km between the two. Spectacular scenery! I did it on a bike a couple of weeks ago. By car/taxi/bus it shouldn't take more than 2 hours - 3 hours max.
MyJordanTrip

From Jordan Jubilee :

From Aqaba to Petra, there are in theory three buses a day, but sometimes there are more. If there is any schedule for this, then I don't know it! The trip usually takes around three hours and costs 3JD. If there doesn't seem to be a bus going to Petra, then have a look for a service taxi, but I rather advise you NOT to accept any offers to drive you from private individuals unless you know them already (questions about insurance and general safety).

i took a public bus (though the other way from petra down to aqaba) and it was less than 2JD and and less than 2hrs. recommend u go early in the morning and ask around:)