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Ben Gurion International Airport is 52km northwest of Jerusalem, just off Rte 1 to Tel Aviv. Airport security is very tight, so if you're flying internationally, check in at least three hours before your flight when flying both to and from Israel.
The airport has good free wi-fi service and plenty of eating options. It is closed for two days on Yom Kippur, which usually falls on mid- to late September.
Nesher Service Taxis operates 24-hour minibuses to Jerusalem from Ben Gurion airport (64NIS per passenger), departing from the rank in front of the international arrivals hall. The minibuses leave when full, so you might have to wait. Tell the driver which part of Jerusalem you're going to and then the driver will decide in what order passengers are dropped off; depending on where your fellow riders are going, it can take a while.
To get to Ben Gurion from Jerusalem, call Nesher a day in advance to schedule your shared taxi pick-up. Service can be gruff, but airport sheruts are reliable and good value.
A private taxi costs 240NIS on weekdays and up to 320NIS on weekends.
GetTaxi, a ride hailing app, works for airport pickups.
Bus 485 (16NIS) goes every hour between Ben Gurion and several stops west of downtown Jerusalem, including the Central Bus Station. It's very comfortable and has outlets for charging (though many don't work) and wi-fi on board. The Jerusalem City Pass includes round-trip travel on this bus. You can't book in advance.