Old Town
Of course you should spend plenty of time in Old Town. Though not as pretty as Prague's, just as mysterious. We arrived on a Thursday. We went out to a very cold Old Town to look for a restaurant, and found a crowded outdoor gathering with mulled wine. This is one example of how Old Town is fun even when it's cold and dark. We then explored the dark streets, and found a couple of watering holes.

Bridge, Bratislava, SK
Church of St. Elisabeth, Bratislava, SK
water roof below a fontain
Old, massive stone buildings in Bratislava