What are some examples of the kinds of foods that would have been eaten then? Would it have been closer to modern Greek or Egyptian cuisine, or something else?
A lot of their food would have recognisable as Egyptian food today. Much would have also been recognisable to Egyptians of Pharaonic times. Dishes such as ful medames (stewed broad beans/fava beans) and molokhiya (green soup of mallow leaves) are considered national dishes for good reason - they have been eaten in Egypt for many thousands of years.
The foundation of the diet was grain (wheat and barley). This was eaten as bread, both flatbreads and thicker raised oven-baked breads, and in other forms such as bulgur wheat and porridges.
Vegetables included onions and leeks, a variety of greens (e.g., mallow, lettuce, radish/turnip greens), okra, cucumbers, a radishes. Legumes were very important in the diet, with broad/fava beans, lentils, and chickpeas all being important (black-eyed beans/peas are common today, but I don't know when these reached the area (they were domesticated in West Africa)). A variety of fruits were available, including dates, grapes, pomegranates, figs, peaches, quinces, and citrons and possibly lemons. And, of course, the olive. Nuts included walnuts, pistachios, pine nuts,
Dairy products such as cheese and butter were eaten. Other animal products in the diet included eggs, a variety of seafood, and meat.
Oils included butter, and sesame, olive, linseed, and rapeseed (as a general term for Brassica oils) oils.
Flavourings/spices/herbs included garlic, cumin, coriander seeds, coriander leaves (cilantro), parsley, and that great ancient Mediterranean favourite, fish sauce (garum etc.).
Alcoholic drinks included beer and wine.
Key differences between their diet and modern food were the widespread use of fish sauce, lower average meat consumption, no rice yet (rice would arrive a few centuries later), and no Columbian Exchange foods from the New World (the chilli and tomato being prominent today, and absent then). Sweeteners were honey and sweet fruits such as dates and raisins; sugar wasn't yet grown in the area (Egypt was a sugar pioneer in the region in the 8th century, and a major producer in the next few centuries). Cooking oil was relatively more expensive, so less deep-frying. And, of course, there were no modern processed pre-packed snack foods.