The above part is a bit awkward logically, especially "for some" and its negation in order to establish the contradiction.Now suppose that there is some such that (is strictly greater than ). But this implies that there are at least distinct elements in the preimage of , contradicting .
My suggestion is to divide B into two subsets for the sake of the contradiction. One of them is the set consisting of and the other is the set consisting of . Use the fact that a covering map is an open map and B is connected, which implies that B cannot be the union of two disjoint nonempty open sets. I'll leave it to you to establish the necessary contradiction.
You can construct a bijective map between for an arbitrary pair of points and in B. For instance, choose a path f between and and lifts that path to g in E whose initial point is and terminal point is such that pg=f.
That seemed like it went way too quick and way too simply relative to the standard exercises in my course. What gives?