Rather say " Good day Professor"

I wouldn't begin a letter to a university admittance committee with "Dear Professor" unless you know the professor's last name. For example, "Dear Professor Smith" sounds ok, but "Dear Professor" sounds a bit odd to me.

I do that alot

It's absolutely fine. *In a University application email.

The best way to start a formal letter to a professor is: Professor Johnson, Then the body here.