It'd take generations of work and billions of dollars for the Sinai to be minimally agriculturally productive and that''s with nuclear power plants working as distillation stills and powering groundwater pumps to bring up up yet more water.
The problem is that deserts, once they form, are very difficult to get rid of.
For grasslands to stick around, you need an annual rainfall of roughly 10" or 25 cm/yr. The Sinai barely gets 5cm on a good year. That's 25cm over 60000 km^2 not including topography, so by my calcs, you'd have to pump 137M L a day to cover that area due to evaporation, leakage, etc.
I'm guessing it would take 20 years and 50 billion dollars' capital investment, not to mention intensive soil culturing and so forth for 20% to be covered by sparse grassland. One thing that would deliver insane amounts of fertilizer would be farming carp, catfish, or other food fish in various water tanks along the distribution network.
In 50 years, and 250 billion dollars capital investment you'd maybe get 50% covered by 45% short grassland, 4% tallgrass prairie, and 1% chaparral with the balance being wet desert without dust storms and slowly accumulating nitrogen, phosphorus, potassium, and other goodies plants need to flourish.
In 100 years and 500 billion dollars total investment, you'd see the Sinai mostly covered with 50% shortgrass prairie, a quarter as much tallgrass prairie, 10% chaparral/scrubland and 1% softwood trees.
By this time, you have enough ground cover to counter the desert heat island effect and for rain to keep dropping on the area to at least keep the short-stem grassland alive year to year, drought or flood. The longstem prairie might have some issues, but with more ground cover, less albedo, therefore, attracting more rain.
After 200 years and 1 trillion dollars, you'd probably see 40% of the area with chaparral covering it, with 45% of it being softwood trees and 5% hardwoods as you saw in Lebanon.
The reason nobody''s really gotten anywhere with de-desertification is because of the titanic amounts of energy needed to treat and move the water, how slowly soil forms, and how nature rarely follows a straight line of forward progress. There's diebacks, there's blooms of one species or another as conditions change and niches appear and disappear in the local ecology. Nature moves at her pace by her rules, not subject to the whims of politics or our collective wishes.
More food for thought...