Can I apply for a Schengen visa for a date earlier than my university invitation letter?

What you need to show for a short-stay visa are plans that look reasonable given your circumstances and the documentation you have.

If the main purpose of your travel is the internship and that itself is well-documented, then it is generally reasonable to combine it with some touristing about since you'll be abroad anyway, and you can certainly present such plans when you apply for the visa. (You'll need to provide an itinerary for that part of your plans, though, just as you would for a standalone tourist trip -- you can't just supply documents for the internship and ask for a longer validity period without further explanation).

If the "tourist" phase is longer that a few weeks, however, then you may encounter doubt whether the internship is actually your main purpose -- especially since you're from Egypt, which is reasonably close to your destination countries, such that "I've flown all the way to Europe anyway" reasoning does not look all that compelling. There is no hard threshold for how much pleasure travel you can tack on, though; it will be up to the consular officer who processes your application.

Here I'm assuming that you're already sure that the kind of "internship" you're traveling for is something Austria allows short-term visitors to do, or that you have secured appropriate permission from the Austrian authorities anyway. (Oftentimes "internships" are something the authorities would consider work, and a visa will not allow you to work).