Filming his run for mayor didn’t work out so well for Anthony Weiner, but New York City Councilman Eric Ulrich is hoping for better luck. Mr. Ulrich, a Queens Republican who is considering challenging Mayor Bill de Blasio, began filming Wednesday for a possible reality show featuring his would-be run for mayor. Surrounded by a bevy of cameras, Mr. Ulrich said he wanted to avoid the fate of Mr. Weiner, a former New York City congressman, whose failed bid for mayor in 2013 was painstakingly chronicled in a feature-length documentary.