The Levinsky Spice market is a must see and do. Besides it being a feast on your senses, it really gives you a feel for the neighbourhood. A lot of the old school shop owners have grown up there and now interact with the young bohemian scene that's crept in. Try out some "Hawaj" spices, mixed dried teas, and of course get yourself some fresh Zaatar.
Levinsky, Tel Aviv-Yafo, Israel