You have plenty of choices to travel to luxor, you may decide the best option for you according to your own time table and budget :
it is most expensive, faster and safest method, prices of tickets can vary from time to time it would be better to contact your travel agent or Egypt Air lines office in Alexandria fore more specific details and availability of seats in a specific date.
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Here is a link to their offices in Alexandria, give them a call
Or you may contact Thomas Cook office, where they can arrange for your entire trip with hotels reservations and taxi to your hotel. Here is a link to their site
Egypt Holidays 2017/2018 | Holidays to Egypt | Thomas Cook
Train is much cheaper and a bit slower, A plane might take you there in one hour, while a train would take you in about 10 hours, there is a Sleeper carriages trains, it is safe as well, you need to check the time and prices in the following link:
Egyptian National Railways
Weather you decide to rent a car and drive it yourself or rent a car with a chauffeur or even with Uber or Careem, this is the least safe trip, and slowest of them all. roads are not safe, it is a long way to drive may be around 1200 KM from Alexandria to Luxor, so many stops, you might need to go to toilet but there is no decent ones on the roads. I would not recommend traveling all that distance in a car ( I have done it my self many times, but roads were much safer than today) , If you decide to take the risk here are the links to get a car service :
A few years ago Hilton Hotels used to have two luxurious boats cruising on the River Nile from Cairo, to Luxor and Aswan in about 3 nights, and fly you back on plane. I guess they have stopped that now. However you may ask your travel agent about the availability of such trip now or even a cruise on the Nile between Luxor and Aswan. This would be your magical trip, whatever your decision is, I wish you all the fun and safety.