Will there be a phone shop open around 4am in Ben-Gurion Airport?
To answer you, I have just checked and the phone shop at Ben Gurion airport (arrivals) will be closed at 4am, however I found a web site for you, for an Israeli company that rents out cell phones and sim cards.you should contact them as they will ship the sim or phone to you for a minimal charge, plus they deliver the phone/sim card anywhere in Israel.