The other answers have shown that cats have been in (now) Israel for thousands of years. The extended claim of British importing cats in the ~1930s to deal with a rat problem appears to be an urban legend:
Local folklore has it that the British are to blame for Israel’s stray cat population, having brought cats in during the period of the Mandate in order to get rid of rats.
Yet the problem could have been created long before that: Geneticists from the National Cancer Institute study established that the world’s 600 million cats are descended from five wildcat matriarchs, including at least one from deserts of Israel, so it may be that the British are not entirely to blame.
Jessica Steinberg, Stray cats and their Tel Aviv strut, The Times of Israel, 2012.
I didn't find mention of the claim prior to 2012 (e.g.
israel british cats, restricted to <=2011, does not give any relevant hits). I also didn't find mention of a rat problem in (now) Israel circa 1930 (e.g.
palestine rat problem 1930 doesn't come up with relevant hits).
The above-mentioned study more specifically states:
The results revealed five genetic clusters, or lineages, of wildcats. Four of these lineages corresponded neatly with four of the known subspecies of wildcat and dwelled in specific places: F. silvestris silvestris in Europe, F. s. bieti in China, F. s. ornata in Central Asia and F. s. cafra in southern Africa. The fifth lineage, however, included not only the fifth known subspecies of wildcat—F. s. lybica in the Middle East—but also the hundreds of domestic cats that were sampled, including purebred and mixed-breed felines from the U.S., the U.K. and Japan. In fact, genetically, F. s. lybica wildcats collected in remote deserts of Israel, the United Arab Emirates and Saudi Arabia were virtually indistinguishable from domestic cats. That the domestic cats grouped with F. s. lybica alone among wildcats meant that domestic cats arose in a single broad locale, the Middle East, and not in other places where wildcats are native.
Driscoll et al., The Taming of the Cat, Sci Am., 2009 (pdf).
Here's a photo of an African wildcat (F. s. lybica):
Image by Sonelle, at the Johannesburg Zoo; sourced from Felis silvestris lybica, Wikispecies.
Driscoll et al. also mentions a ~3700 year old ivory cat statuette made in (now) Israel, which they conclude "suggests cats were a common sight around human settlements in the Fertile Crescent" (they also mention a ~9000 year old feline molar tooth).
Another news report makes a similar claim:
"Because there are so many cats, it's become so emotional in both ways," says Idit Gunther, a veterinarian and lecturer in veterinary medicine at Hebrew University of Jerusalem. ...
Many tour guides and Jerusalem residents blame the British, who ruled the city between 1917 and 1948, for bringing cats to the city via their ships. But the felines have lived in Jerusalem for thousands of years, Gunther says. In fact, the genetic makeup of most of Israel's street cats shows they are directly descended from the original African cat domesticated by the ancient Egyptians, and do not include genes of any wild European cat species, she says.
Sara Toth Stub, Jerusalem's Controversial Cats, USA Today, 2019.
Israeli zoologists offer the following reasons for Israel's current cat problems:
There are several reasons for this. First, in Israel organic waste is often not properly treated. Food remains are found everywhere near garbage containers, in yards and parks, readily available for cats. Second, feeding stray cats is a widespread phenomenon in Israeli towns and villages. Individuals providing food for dozens and even close to a hundred cats are not uncommon. Third, Israel has a mild climate and regularly-fed cats easily survive the winter.
Brickner-Braun, Geffen, Yom-Tov, The Domestic Cat as a Predator of Israeli Wildlife, Israeli Journal of Ecology and Evolution, 2007. (pdf)