Bagenal's Castle is now home to Newry and Mourne Museum, which means you get the Tudor stronghold and a free local museum in one visit. The collection covers the town's monastic origins, the canal era, the linen trade, and the more recent Troubles, with some surprisingly good displays.
The building itself is the real draw. The Tudor staircase, fireplaces, and gun-loops are unusually intact for Ireland, partly because of the centuries spent walled inside a later structure. The museum is on McCann's Bagel Bakery's old site on Castle Street and is normally open Tuesday to Saturday.