How did the Ottoman Empire pass the Sinai desert?

The decisive battle against the Mamluks had arguably already been fought at Marj Dabiq near Aleppo a few months prior to Ridaniya. At Marj Dabiq, a force of some 20,000 Mamluk troops was defeated with relative ease by an Ottoman force about three times as large. The Mamluk Sultan Al-Ghawri was killed and a new Sultan had to be selected in Cairo. The Mamluk army more or less scattered.

After an initial period of uncertainty, Selim decided that it was necessary to conquer Egypt as well as Greater Syria. To that end he sent a force against Gaza, which has historically been the jumping off point for invasions of Egypt, or the focal point of the defense of Syria during invasions from Egypt (for example, the First World War battles of Gaza.)

As for the crossing of Sinai itself, there are several routes through Sinai by which a large force can march through the desert with intermittent stops at desert wells and in a mostly self-sufficient manner. That, for example, was the method used during the 1915 Ottoman Raid on the Suez Canal, which took the following routes: http://3.bp.blogspot.com/-bqhCJI4_QsY/VH_IaIsJoPI/AAAAAAAAJ3k/SKM_eldjJqs/s1600/17_suez-attack.jpg

While bedouin raiding of the army was a potential concern, the desert itself was not defended by the new Mamluk Sultan, and "the main body of the army arrived without difficulty in Egypt, and on...January 22 the Ottoman vanguard reached Birakat al-Hajj, a few miles north of Cairo.

It's perhaps also worth noting that this was done in the winter, when temperatures are mild. In Arish this week (February 15, 2018) for instance, which is one of the largest cities in north Sinai, the expected high temperature is only 24C with a low of 9C (75F/48F respectively.)

Some Turkish historians says we had nearly 300 k army is that true?

My source for the above is the Cambridge History of Egypt, Volume One, Islamic Egypt, 640-1517 and doesn't mention the size of Selim's army during the crossing of the Sinai, but as I mentioned above it does mention the army's size at Marj Dabiq as being about 60,000, which the author comments as being exceptionally large.

I would find an army of 300,000 to be incredibly unlikely under the circumstances. An army of far less than 60,000 for that expedition seems more likely given that the same force also had to occupy newly-conquered Greater Syria.

Won the Battle of Ridaniya

As I mentioned, the "real" battle was Marj Dabiq in the months before. The Battle of Ridaniya was won in only 20 minutes. For one thing, the Ottomans were able to attack the largest Mamluk guns from the rear, rendering them useless. Ottoman skill with firearms generally rendered the Mamluk fighters basically outclassed. And although the Egyptians managed to field about 20,000 men, these were composed in large part of convicts, bedouin, and lower-class civilians who had been scraped together to bolster what was left of the army after Marj Dabiq.

Source wise As I said, I'm mostly going by the account in the chapter "The Ottoman occupation* in the first volume of the Cambridge History of Egypt.

edit: "this week" is 2018, not 2017, doh. :)