Cairo has always been a great place to shop for traditional items because of its spectacular medieval marketplace, the Khan al-Khalili, where browsing and bargaining are half the fun. There is no tried and tested bargaining strategy; just shop around, decide how much something is worth to you, and start bargaining lower than that in order to end up at that point. In the Khan, the opening price is never the final price.
As for modern or more-practical items, until not long ago Cairo had almost Soviet-style limitations, but that has changed dramatically. There is no particular shopping district (although Mohandiseen comes close), but the World Trade Center in Boulaq has more than 100 shops and is perhaps the best single source.
Concerning antiques, although most of what you see are reproductions, Cairo has a long tradition of connoisseurship in collectibles and there is always the possibility of finding a real gem. Be prepared, however, for local tastes that favor ornate French-style furniture and antiques, not the Middle Eastern pieces you might be longing for. There are several nameless antiques shops along Shara Hoda Sharaawi (Downtown) that are worth looking into.
For a surprisingly extensive selection of foreign-language newspapers and magazines, try any major hotel bookshop or the cluster of stands at the corner of Shara Hassan Sabri and Shara 26 Yulyu in Zamalek. Alternatively, try the stand Downtown next to Groppi's on Maydan Talaat Harb, or the one near the McDonald's across from the entrance to the American University in Cairo.
International clothing companies like Mexx, Daniel Hechter, Naf Naf, and Benetton manufacture in Egypt, so they have shops throughout the city; there is also a row of them Downtown on Baehler Passage. You can also find local brands like Mix & Match, On Safari, and Concrete. Generally, though, this is not the place to purchase famous designer clothes.
Stores open around 9 or 10 and close, depending on their location and trade, anywhere from 6 to midnight. They generally close on Sunday if they close at all, but most are open seven days a week.