It's actually possible to escape the crowds and the noise in Cairo, although you have to put up with a lot of both on the way. I would recommend Beit el-Suhaymi, a wonderful, labyrinthine Islamic house-turned-museum where you can really picture how the large families used to live.
Before or afterwards take a stroll around the north of Khan al-Khalili market. Away from the hassle of the market stalls you see a bit of real innercity life. I am female and, being there on my own, I didn't feel hassled at all in this part. Be sure to respect their dress code though.
Another tip is go to the Mosque of Sulayman Pasha when at the Citadel. Around the Citadel itself, in particular the Muhammad Ali mosque, was very busy, but the simpler, smaller mosque felt like a peaceful oasis and has lovely mosaics.
Beit el-Suhaymi, Khan al-Khalili market and the Citadel are all located within the city centre.
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