Mr. Lynton, who weathered various corporate crises at Sony Entertainment over his 13-year tenure, including a devastating cyberattack and a battle with activist investors, will step down on Feb. 2, the company said. After that, Kazuo Hirai, the president and chief executive of Sony Corporation, based in Tokyo, will take a more hands-on role at the conglomerate's movie, television and music division, at least for a time.