Things to See:

Old City (All 4 Quarters)
Western Wall (Wailing Wall)
Church of the Holy Sepulcre
Via Dolorosa
Dome of the Rock
Temple Mount
Mount of Olives
Machane Yehuda Market (The Shuk)
Yad Vashem (Holocaust Museum)
Dead Sea (An hour drive)
Masada (An hour drive)

Restaurants:

Hamarush 

Sabich Aricha

Azura

Ishtabach

Piccolino 

Bars:

Machane Yehuda Market (walk around here at night for lots of bars, hookah, etc)

Gatsby Cocktail Room

Mirror Bar

The Garden Terrace (Waldorf Astoria)

Rooftop Bars:

Rooftop (Mamilla Hotel)

Lodging:

King David Hotel
Abraham Hostel Jerusalem