One of the things you must do is go to (and through) the main gate of the old city. Beyond the gate [which used to be closed for the night up to the 1950's] you enter a different world. A maze of streets and alleys where all conceivable shops are located and handicrafts are made and sold on the spot. As tourists are compartively rare, and Western tourists almost absent, most trade is local business.
The Palace at Dar al Hager is a short trip out of Sana and well worth the effort. A travel permit is not required and you are allowed to use public transport. A shared taxi costs 50YR to the closest town and from there you can hitch or walk the remaining 3kms. A taxi especial will take you all the way but costs about 10 times more. I came back on a local bus for 10YR.
The palace entrance cost 500YR.