Serendipity, you have a few optionsEgyptair (to Dubai):
they have 7x weekly flights from Cairo
they have 2x weekly flights from Alexandria
they have 1x weekly flight from Sharm El Sheikh
Emirates (to Dubai):
they have 8x weekly flights from Cairo
they have 7x weekly flights from Alexandria
Singapore Airlines (to Dubai):
this airline has 3x weekly flights between Cairo and Dubai. The flight route Cairo-Dubai-Singapore but they offer the Cairo-Dubai sector for sale,
Air Arabia (to Sharjah):
they have 14x weekly flights from Alexandria
they have 2x weekly flights from Assuit
they have 2x weekly flights from Sharm El Sheikh
they have 2x weekly flights from Luxor
**as mentioned aboce Sharjah is a 40 minute coach ride to Dubai
And yes it would be cheaper to book flights through a travel agency or airline office in Egypt. If you want to travel from Cairo, then you can expect to pay between 200 to 600 Euros for a return ticket depending how early you book your flights and when you want to fly.
Hope this helps and feel free to ask more questions.