午前八時 doesn't have any other usages than the "time location" even without に. It's neither just "eight" nor "eight hours". 午前八時に is an adverbial phrase, so it can't be the subject, object, or predicate with copula. You can think 私の出発は午前八時です as 私の出発時刻は午前八時です.
As Elizabeth-san pointed out, 午前八時には私の出発です sounds awkward, but 午前八時に出発です is OK. In the sentence 午前八時に出発です, the predicate 出発です is more close to a verb 出発する, that's why に is used here. In fact, you can use 午前八時の出発です when 出発 is used as a noun.