Take a train to Central Station (Brussel Centraal / Bruxelles Central) and walk to the main square (Grote Markt / Grand Place). Have a drink in one of the bars, then walk over to the Royal Galleries (Koninginegallerij / Gallerie de la Reine) and have dinner at Taverne du Passage (and not in the tourist traps in Rue des Bouchers).
Book a hotel near the central station/main square (Novotel, Best Western Premier, Ibis, Le Dixseptième) or the Sheraton at the airport. The other airport hotels require a shuttle and you'll lose time. In the morning, catch a train back to the airport.