Elvira Nabiullina spent eight years modernizing Russian monetary policy and forging bonds with global markets. In the past six weeks, she took apart much of what she created for a new task: keeping President Vladimir Putin ’s war economy humming. Ms. Nabiullina, governor of the Russian Central Bank, is among the key players in Moscow’s efforts to prevent Western sanctions over the Ukraine invasion from causing an economic meltdown. An ally of Mr. Putin for two decades, she is the primary official responsible for stabilizing the ruble and combating inflation—tasks the Kremlin sees as essential in shielding Russia’s population from the fallout of sanctions.