Manhattan apartment sales fell sharply and prices slipped in the fourth quarter, but brokers remained undaunted and predicted that a boost in buyers’ confidence would stabilize the market in 2012.Manhattan sales declined 12.4% in the fourth quarter from the year-earlier period, falling to the lowest level since the summer of 2009, according to a market report to be released Wednesday by Prudential Douglas Elliman. Sales were off more than 35% from strong sales in the third quarter.