It sounds like you have your trip pretty much planned out already. In answer to your questions:
Whether you go directly to Eilat or travel via Masada is a matter of preference, as well as determined by what mode of transportation you use. You can get to Eilat from Tel Aviv by car, bus, or plane. If you fly to Eilat, then obviously, you will have to get to Masada another way. Flying is the fastest way to Eilat. A bus will be a long bus ride, while driving south will afford you the greatest flexibility in terms of route and scheduling.
Currently, there are no day tours to Masada that allow you to be there for the sunrise. The Masada national park closes at 5 pm (summer hours) so you would not be able to see the sunset from the top of Masada either.
If you chose to stop at Masada on the way to Eilat, you would need to drive, either in a rental car or taxi, or by hiring a licensed tour guide who has a touring vehicle. You would then have the option of going through Jerusalem and approach Masada from the Dead Sea (east) affording you the option of climbing the snake path or taking the cable car to the top. The other route would take you through Arad, approaching Masada from the west, where you would have to climb the ramp up to Masada. You would then continue the drive down to Eilat after your visit at Masada. A similar option is available if you fly to Eilat and then drive north to Masada.
Make sure that which ever route you pick you have plenty of water, a hat, and sunscreen - it is very hot and you need to keep yourself hydrated.
Whatever you decide, it is sure to be a memorable experience.