Bur Dubai is the best spot to get a feel for how the city felt and looked before the huge increase in construction that took place in the 90′s.
Textile Souk: visit the textile souks and barter with locals for clothing, scarves and little bit of everything.
Bastakiya: this area of Bur Dubai is designed to have an older feel and the look of traditional buildings, however, because it is so incredibly new you feel a little as though you are in a theme park. It is still very worthwhile to visit. Take a stroll around and visit some of the art galleries and coffee shops – you can also stop and get a camel burger while you’re at it.
From Deira you can take an Abra ( similar to a water taxi ) for 1 Dirham to cross the creek to Bur Dubai. The journey only takes about 5 minutes and is extremely convenient.