Go to the Egyptian Museum in the morning to avoid crowds
The best time to go to avoid crowds is right after museum opening (9am) until noon
The museum gets more crowded in the afternoon, as group tours start to show up en masse (Chinese and European tour groups mainly)
However, to put it in perspective, I went in the afternoon, but the crowd wasn't THAT bad. Egyptian tourism is still down compared to its peak many years ago I think. Even though the Egyptian Museum is one of the top attractions in Cairo it's nowhere near as crowded as some of the places I've been to in Europe