I travelled from Israel to Egypt on a tour bus about 7 years ago. I'm sure that being with this kind of group greased the skids a bit, but it was still a bit tense.

We got out of our nice mercedes bus on the Israel side, carried our luggage maybe 50 yards across a kind of no-man's land (guns, fences, no photos allowed military barricade barbed wire kinds of stuff) to the Egypt side. Much dingier building, lousy bus, hot & dusty. I think it took us 1-2 hours. All luggage & packages had to be gone through. I observed some young people who looked like they had been travelling a while on the cheap--it looked like it was taking them all day.

The coolest part of the trip was crossing the Suez canal at sunset, and seeing the gun posts all around us.

The passport issue is real. If you do enter Israel, ask them to stamp a blank piece of paper you have placed in your passport. A number of countries do not recognize Israel and may be passively hostile about the passport. Sad, but true.

Israel and Egypt are wonderful countries to visit, and I'm hearing the current turmoil is creating good deals for tourists. As one who has made a number of solo trips overseas,(and as one who despises organized tour groups) this is one trip I would remain more comfortable doing with an organized group, or having a personal contact in the country to act as a "handler". Some tour groups may be willing to work with your miles. You might have luck checking with universities, churches, and synagogues to see if they have an upcoming trip you can piggy-back on.

All this said, the crossing between the two countries can be done, but is likely best done with a commercial operator that understands how to get through the red tape.

Good luck! Let us know what you come up with.