You will be perfectly fine. Taxis line up outside the door so just collect your luggage and firmly ignore any offers of help until you get to the official taxi line. You should expect to pay around 50-70 Egyptian pounds to get to your hotel, less if it's one of the ones near the airport. Depending on the traffic, it should take around 45 minutes to get to the downtown area.

Another option is to arrange for a "meet and greet" at the airport -- this will be someone who will meet you before immigration, take your passport and the money for the visa and then handle all the arrangements for you before whisking you through the line, picking up your baggage and driving you to your hotel in a car (or possibly a van). If you're working with a travel company in Egypt for the land tour, they can arrange this for a separate fee (seems like I paid around $30 US last time, but I can't remember). Any large hotel should be able to arrange this as well.

"Muslim standards of modesty" differ quite a lot according to the Muslim, incidentally. In Cairo, you'll be fine in slacks and a long-sleeved shirt -- no need to break out the uber-conservative wear until you hit southern Egypt, though having a headscarf on hand is always a good idea for visiting mosques.