The drive from TLV to Eilat takes over four hours (at what hour does your flight arrive?). Alternatively you can fly directly from TLV to Eilat: look at the Arkia and Israir websites for schedules and fares.
If you drive down there is a lot to see along the way. You may want to stay overnight in Mitzpe Ramon or the Dead Sea, for example.
Will you have the car for the entire week in Eilat? You can drive to Timna and Yotvata.
You can also take a day tour to Petra (but you can't take the car across the border).