How necessary or exaggerated is the following Anrede?
"Sehr geehrter Herr Prof. X" bzw. "Sehr geehrte Frau Dr. Z"
I mean, should I write Herr Prof. X, or is Prof. X enough?
Also, is it old-fashioned to write like this?
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First, it is not correct to leave out the Herr or Frau. “Sehr geehrter Professor X” sounds incomplete.
It is possible to leave out the academic title (Professor, Dr. – the general recommendation is not to abbreviate the former in the salutation), but it depends heavily on the context whether it is advisable. Generally, the need for including the Professor increases the older and the more conservative the recipient is, the more hierarchical their field of study (think medicine), and the more distant your relationship to them.
There are professors who want to be addressed “Herr Professor Dr. X” at all times, even when spoken to. There are others who would consider this exaggerated.
If in doubt when writing, do include the title; it is safer. At a personal encounter, listen to what other people with a similar status use; and again, it is safer to include the title if in doubt. (“Similar status” because it is not unusual that professors who expect the title from common people are OK to be addressed without it by fellow professors.)
There is some confusion in the present answers particularly with regards to the technicalities of the salutation. Actually the whole subject is pretty straight forward and how to properly salutate a professor can be googled. In the salutation (other rules apply for, e.g., for signatures or letterheads), you will say "Sehr geehrter Herr/Frau Professor/Professorin" until you know that you can leave out the Professor/Professorin. You don't include "Dr." as soon as someone is a professor. You don't abbreviate "Professor" (i.e. not "Prof.") and you don't spell "Dr." in full (not "Doktor"). Some authors argue that "Professor" is a "Dienstbezeichnung" and thus consumes the "Herr/Frau" as it consumes the "Dr.". So leaving "Herr/Frau" out is not wrong, maybe even theoretically required, but not very widespread.
Now to the question whether you actually should include the correct salutation: From vast experience I can assure you to always use it (correctly of course) – especially if you're unsure whether you have to use it; it is not old fashioned at all and most still expect it.
According to what my teacher told me,
Sehr geehrter Herr should be there followed by the name. You should only use Prof. if the person himself referred to his own name with the academic title(In other words, if you have evidence that the person prefers that he be called with the academic title).