To unlock Old Town in Dying Light, you must first unlock "The Saviors" mission. To travel to Old Town, interact with the poster depicting the Ember's Tower. To travel back to The Slums, interact with the poster within the Ember's Tower depicting The Slums. This process is the same as initiating The Bozak Horde DLC and traveling to the Harran Stadium. Note that you cannot fast travel between safehouses in Dying Light.