No idea about cheap hotels, but I heard the youth hostel is quite nice.
What to do? -Depends on what you like?
Museums: Universum-highly recommended, a lot of self-action in there and not at all boring
Go out: Schlachte, Viertel, Pubs
Take a stroll through Viertel with many cafes etc.
If you are interested in beer, you might want to take a tour through Becks brewery?! Guide english tours available.
Hope that helps you out.
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