I have done some of this.
The Eilat to Jarusalem stretch was okay up to Ein Gedi. I stayed there and paid to have the bike put in a van from Ein Gedi up the hill to Jerusalem - I saw 45'C that day.
The road from Ein Gedi was narrow and a fence on the right side with not much area to escape. The traffic was not moving slowly either, so I was glad to be in the van.
The Road from Eilat to Ein Gedi was fine not the most interesting but easy enough even in the hot weather - I think it is actually downhill as the Dead Sea is actually below sea level Eilat !
The best parts of my tour were from Jerusalem to Tel-Aviv through some of the mountain tracks. Also the best was Ha Basor to Mitzpe Ramon and on to Eilat. Fantastic route the Crater at Mitzpe I did not even know it was there until I saw it !
Some pictures
The back road past the tank training ground,
Between Eilat and Ein Gedi
Approaching junction from Mitzpe to the Eilat Road, overlooking Jordan Valley
Overlooking the crater at Mitzpe Ramon - I had to get out of its way !
The back woods out of Jerusalem heading to Tel-Aviv
I loved it, had a great trip.
EDIT:
Regarding your comment of cycling the bit past the Dead Sea. For me it became a bit of a trap. The day before there had been a thunderstorm nearby and a little rain, which was glorious, the smell of the desert with a little rain. But it was cooler and there was some cloud, so I cycled on to Ein Gedi. The next day it was back to baking hot, but worse because the Dead Sea seemed like a big sun trap. So I had come into it in the cool, and the sun came out again into a furnace that did not relent. So paid for a van ride from Ein Gedi. The road was narrow, twisty, with traffic and in places the crash barrier very close. I have no regrets about travelling that section in a van. I saw the temperature reach 45' C !
I cycled from Ben Gurion - Ashkelon - Sderot - Ein HaBasor - Mitzpe Ramon - Eilat - Ein Gedi Jerusalem - Tel-Aviv
