Check out Bilu Bikes and Pedalim.
There’s another one that’s Merida-affiliated just south of Baka towards Teddy stadium, but I can’t seem to find them online. The owner was super helpful and friendly when I went there to race GFNY and watch the Giro a few years back.
As you’d know having been there before, it’s an AMAZING place to ride. I’ve ridden in France, Italy, Australian high country - the climbs around Jerusalem are right up there among the best.
Carry salt tablets. Again, as you’d know, it’s a desert. At some stage you almost certainly WILL cramp no matter how disciplined you are with hydration prior.
I’d recommend not taking your bike. I took mine from Australia and the handling at the airport really left a lot to be desired. Solid case had split in two at the bottom. How they managed that I have no idea. Bike was fine thankfully, but I ride metal bikes… I’m pretty sure a carbon bike would have cracked.
As for riding groups, the sport has really taken off around there in the last few years (Giro, ICA, ISN) but I’d suggest just hitting the climbs on a Sat/Sun morning and latch on to a paceline. Scheduled rides and proper group organisation in Israel - not so much 
Sat mornings will be blissfully quiet as it’s the Sabbath and there’s barely any cars on the road.
That GFNY ride pre-Giro is honestly my equal first most epic ride ever done and likely will do. Smashing around Jerusalem on closed roads was absolutely epic.
Have fun!