Ben-Gurion to Eilat by (rental) car

Hi, I am planning to have a week long visit to Israel early in January. My wife wants to spend three to four days in Eilat by the sea, and we plan to stay in Jerusalem rest of the time. Please can you advise whether it is a good option to have a car rental from Ben Gurion airport where we arrive to, and drive to Eilat from there? And is the road 90 the best option (on the map there are also other road runing to the west of this one). And am I right to think that we need to drive through Jerusalem anyway before we hit the road 90? How long does it take and is it ok to drive outside daylight (I guess we only arrive around 4pm to Ben Gurion so will need to drive in dark) - is it save?

Driving in the dark will be safe through either route.
option one
from Ben Gurion take route 1 to route 443 to Jerusalem. about 2km past the main Jerusalem checkpoint take the exit marked 'Ma'ale Adumim'. keep left off exit. left at next traffic light. Left again at next traffic light. This will take you onto Route 1 out of Jerusalem to the Dead Sea - to Route 90 which goes all the way to Eilat. It's a great drive. You'll catch the sunrise en-route and pause at the Dead Sea - En Gedi is a good place to have breakfast near the sea. (4-5 hours driving)
Option two
From Ben Gurion take route 1 - to route 6 south - to route 40 south. Then you can either take route 31 via Arad to the bottom of the Dead Sea and turn right on route 90 south to Eilat.
OR
Take route 25 which also comes to route 90, just a bit further south.
OR
continue on route 40 past Be'er Sheva and Mitzpeh Ramon. It eventually links with route 90 much further south.

I suggest taking option one going south to Eilat and option two when you return to Jerusalem.
Also on the return, during the DAY ONLY, you can safely drive through the West bank to Jerusalem via route 60 north from route 31. That way you will see lots of Israeli settlements and Palestinian villages. Route 60 bypasses Bethlehem to the west, entering South Jerusalem