I am currently a Hill staffer and have been for about two years. Grad school prestige, in my opinion, does not matter. Like others have said, you need to start at the bottom, regarrdless of where you went to school. I know people who have MAs or MPPs from good schools, and still started as Staff Assistants (the entry-level job on the Hill) and worked their way up. I would recommend interning in DC between your first and second years of school if you really want to be on the Hill (also I wouldn't commit to a career here until you have tried it out!). It will be easier to get an internship during other seasons, but I wouldn't decide to go to school in DC for this reason alone if a program somewhere else is a better fit.
As a Staff Assistant I made barely over $30,000 and now I make just over $40,000. The pay and hours are rough, so I would definitely recommend spending some time on the Hill and making connections before committing to a career here.