Was Manfred a protagonist or antagonist? Why?

The two terms originated in ancient Greek tragedy. They are both derived from the word "agon," which means struggle or contest. Thus the "agonists" were contestants or opponents. Originally, tragedy was performed by a chorus with a chorus leader. The chorus leader eventually evolved into the first individual actor or "protagonist." Aeschylus added the second actor (deuteragonist), and Sophocles introduced the third (tritagonist). In general, the most skilled actor took the role of "protagonist," the leading character with the best role and most lines, and another skilled actor would take the role of deuteragonist, a character who was generally engaged in some sort of struggle or contest with the protagonist and thus might be called an antagonist. In the classical sense, therefore, the term "protagonist" means something like "starring or leading role," while the antagonist is a character who opposes the protagonist.
In The Castle of Otranto by Horace Walpole, Manfred is perhaps best classified as the protagonist, as he is the central character—even though he's also a villain. Isabella, Theodore, and Father Jerome, the more sympathetic characters, do not really take a starring role in the novel, each being treated in fewer pages than Manfred.
Sometimes literary terms developed for discussing certain works do not describe the structures of other types of works very effectively and thus should not be applied. It might be best simply to call Manfred a villain; he fills this role in a manner quite typical of the Gothic genre.