At TLV, stay airside—security is intensive, so relax, eat, and stay close to your gate.
Take the train from Ben Gurion Airport to Tel Aviv HaHagana or Savidor-Merkaz in 15 to 20 minutes (ILS 17 / $4.60). From Savidor, take a 10-minute taxi (ILS 30 to 40) to Carmel Market or the Rothschild Boulevard area. The market is open Sunday to Friday, 8 AM to sunset. Allow 75 to 80 minutes total for the return journey and give yourself 4 hours before your departure for Ben Gurion's security process.
Old Jaffa (Yafo) is 15 minutes by bus from central Tel Aviv (Bus 10 or Grab, ILS 20 to 30). The old port area has an ancient flea market, the Jaffa Clock Tower, and strong hummus restaurants. Hummus Abu Hassan on HaMifats Street (from ILS 42 / $11) is a local reference point: arrive by noon or expect a queue. Add a walk north on the Tel Aviv beachfront promenade (4 km of public beaches) before heading back.
Jerusalem is 45 to 60 minutes by train from Ben Gurion (change at Tel Aviv HaHagana, ILS 31 / $8.50 total). The Old City is 15 minutes by taxi from Jerusalem station. The Muslim Quarter, Western Wall, and Via Dolorosa are all within walking distance. Allow 3 to 4 hours in Jerusalem and 2 hours for return transport. Factor in 4 hours for Ben Gurion departure security.