In Egypt whole families apparently manage to survive on an income of about 300 LE (Egyptian pounds, about 50 USD) per month.
I've looked into living in Cairo for a year in the past, and from the data I gathered in 2005 it seems to be possible to live comfortably on around 200-300 USD per month in Cairo. (This does not include any fees you might have to pay to the American University of Cairo - AUoC.)
In Egypt everything has 2 prices, Egyptian prices and tourist prices. The big majority of Egyptians will add a hefty surcharge when dealing with a foreigner. Say a shared taxi ride that is 1 pound - you as a foreigner are bound to be charged 5 pounds if you are lucky and look like you know what you do or 200 pounds if you trust your taxi driver to charge you correctly.
In 2005 the campus of the AUoC was near Midan Tahrir, which is the main area for hostels in Cairo. There where plenty of students who resided in the hostels while looking for a place to stay, I think most of them ended up paying rent that was around 50-100 USD per month. (That is tourist price - practically unavoidable if you are a westerner and need something that let out for a year, not for life.)
Cheapest way to have lessons is to watch Egyptian TV. There should also be plenty of Egyptian students willing to earn a few pounds, which shouldn't be too expensive. As I wrote, whole families need to survive on 300 LE per month, so paying somebody 20 LE per hour for lessons seems resonable to me from my experience of Egyptian prices. Basic previous knowledge of Arabian grammar and lexic is assumed.