There is no night bus on Friday night/Saturday morning. If you come back from Eilat by bus, it means you have to go to Eilat by bus, unless you return your rental car in Eilat.
Chatul, OP says they are arriving in Tel Aviv, presumably meaning their plane lands at Ben-Gurion Airport, at 17:20 on Friday. There is no way to get from Tel Aviv (or the airport) to Eilat by public transport on a Friday night.
The first bus from Tel Aviv to Eilat on Saturday is at 16:00, arriving in Eilat at 21:39.
OP says:
>Me and other 2 friends are travelling to Tel Aviv for work. We got this idea to visit
Petra while we are there.
But then goes on to say
>We arrive to TLV @ 17:20 on Friday 12th; we will rent a car and we will head
to Elait.
This isn't at all clear to me. OP, you will be spending some time in Tel Aviv working, and in addition you want to go to Petra. But as I understand it, you are coming straight off the plane and heading that same night, Friday night, to Eilat? That doesn't sound like a very practical idea to me. Your profile says you are from the Netherlands Antilles. Is that where you'll be flying from? If so, you'll be jetlagged, and you're planning to go straight on this long and boring drive, on a dangerous road at night.
I would spend at least a day in Tel Aviv first, just to get your breath back, and then head to Eilat and Petra.