Things to do in Sanaa
Sana'a is a great place for people-watching and culture observing. The unhurried pace of Yemeni life, coupled with the almost total lack of industrialisation seen in nearby countries, means that it's quite possible to pull up a chair in a coffee shop and watch the world go by for hours on end.
If you want an experience you can only get in Yemen (and are prepared to risk your health): participate in a qat session. Qat (khat) is a mild narcotic chewed by most males in Yemen. They spend their afternoons hanging out, chewing the leaf, and conversing. A bag will cost no more than 5 dollars, and you can get "inferior" product for a cheaper price. Side effects are lack of appetite and insomnia. Qat is considered by many as the prime cause of Yemen ranking as the poorest country in the Middle East. Note that Qat is considered a controlled substance in the United States and other countries (i.e. it is illegal and those trying to bring it into such countries risk arrest).