Is it easy to get around there?
Yes.
Public transport is excellent. There are the local buses in the cities and the Egged buses are running all over the country. It helps if you know enough Hebrew to identify destinations on buses, but it is not really necessary.
Apart from buses you got sheruts, shared taxis. These are minibuses that run on certain routes, they are faster, more frequent and get you to out of the way places, but they are also more expensive and their drivers can screw you over. Always know the price before you get in, a good guidebook will list them. If possible make sure you know the price of the Egged bus ticket so when the sherut driver tells you a price you can compare it.
How much would I be looking to spend daily... Staying in hostels, eating basic food etc.
This info is a bit dated, but I think you are looking at around 70 AUD per day and person, more if you like eating out or partying.
And how easy is the transport to Cairo?
Transport to Cairo involves a bit of leg work. It is not very difficult, but takes some planning. First you need to get an Egyptian visa either at the embassy in Tel Aviv or at the consulate in Eilat. Then you need to find out when the buses run, IIRC there is only one bus daily from Taba going to Sharm el-Sheikh. Don't plan on going from Jerusalem or Tel Aviv to Cairo in one go, break up the journey either in Eilat or the Sinai or both before going to Cairo.
http://www.ask-aladdin.com/bus_schedules.htm
Tip: If you are going to Israel for religious reasons, definitely stay for a few days in the Sinai and visit St. Catherine's.
http://looklex.com/egypt/st_catherine.htm
http://www.ask-aladdin.com/stcatherine.html
From Sharm e-Sheikh you can go to Cairo by bus or take the ferry to Hurghada. If you plan to tour Egypt it would make more sense to take the ferry to Hurghada and continue from Hurghada to Aswan, visiting Abu Simbel and then making your way down the Nile to Cairo.
Get some decent guidebooks to help you with your planning. I like the Lonely Planet, almost everybody uses it for Egypt. I also had the Lonely Planet when I went to Israel, but this was 10 years ago and I don't know how useful the current edition is. Two or three years back at the height of the internal troubles in Israel the Bradt guide was the only decent guide book for Israel.