That tour spends only part of one day in Petra. It also spends two days in Aqaba, which I don't think is worth the time (I'd reverse those two or skip Aqaba entirely in favor of touring Amman, more time in Petra/Wadi Rum, or seeing any number of other places in Jordan), and lists three nights in Dubai, which you probably have pretty well under control already, though I don't see any activities listed for them.
However, a quick Google search for Petra tours from Queen Alia Airport (AMM) turned up this one:
https://www.memphistours.com/Jordan/...-Amman-Airport
I have no interest in or experience with this company, so this is not a recommendation - but if I could find it in about fifteen seconds, someone who is more motivated to check out such tours ought to be able to find others and read their reviews. Bottom line is that what you're looking for does exist.