The popular discontent that engulfed Republicans amid the partial government shutdown has now washed over President Barack Obama, whose job approval rating has sunk to an all-time low in a new Wall Street Journal/NBC News poll.

Americans just weeks ago heaped scorn largely on congressional Republicans over the dysfunction in Washington. But the new poll found a sharp turn against Mr. Obama, during a month in which lawmakers tiptoed up to a potential debt default and the White House fumbled the rollout of its signature health-care...