Graded ideals over the polynomial ring are studied deeply with a huge of methods and results. Over the exterior algebra, there are not much known about the structures of minimal graded resolutions, Gröbner fans of graded ideals or the Koszul property of algebras defined by graded ideals. We study componentwise linearity, linear resolutions of graded ideals as well as universally, initially and strongly Koszul properties of graded algebras defined by a graded ideals over the exterior algebra. After that, we apply our results to Orlik-Solomon ideals of hyperplane arrangements and show in which way the exterior algebra is useful in the study of related combinatorial objects.