Theoretically possible, perhaps, but I would not recommend it. It's a long drive from TLV to Petra via Jerusalem & the Allenby Bridge (I'd guess 4 hours one-way minimum plus border hassles), and flying to AMM and driving from there won't shave off more than an hour.

Once there, Petra is too large to be seen in a day even if you're staying there, never mind just a half-day -- for example, if on foot, it's a good three hours (half of that uphill) just from the entrance to the Monastery. Camels and donkeys can save you the effort, but won't speed you up much, and cars are not allowed inside.