I've got a couple of questions for you before being able to offer much advice.
What is the purpose of your visit? Tourism? Work? Study?
How long will you stay? You say you want a month to month lease, but you will pay significantly more if you come like that without at least a 6 month stay and willing to sign a 6 month lease. Is 6 months really too long? Short term rentals can be as much as 2 or 3 times the rent...and if 6 months were probably the amount you are thinking to stay anyway, it might be worth it to sign the 6 month lease to keep the rent down and if you only stay 3 months and had to pay for the additional 3, you'd loose nothing... See my point? They prefer one year leases to really give you "long term" status, but you might be able to pull it off with 6 months.
For a start, I'll say that probably areas like Mohandiseen, Dokki, Garden City and Zamalek are your best bets for location. Another area that has lots of foreigners in it (if that is what you want) is Maadi, but that in general is overpriced, or there is Heliopolis, which has fewer foreigners, but is a safe decent neighborhood. Both of these are a bit out of the way though if your reason for being here is purely tourism.
Edar is probably the most widely used realestate company on line here. try edar.com or go to google and type in edar and see if you find it.
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