You might want to spend some time browsing the exhibits at Museum of the World Ocean, Bunker Museum, or Amber Museum. Ruins of Koenigsberg castle, King's Gate, and Wrangel Tower are monuments to see when you're exploring the area around Cosmonaut Monument. You can watch performances at Kaliningrad Regional Drama Theatre, Queen Louise Kirsch Puppet Theater, and Kaliningrad Regional Youth Theater for Young Spectators if you're interested in local theater.