It will be more important to know "whether Obama will demonstrate more willingness - more bravery, actually - to take on issues he ducked the first time around", the Post said in an editorial.
These, it said, included reforming entitlements, particularly Medicare, and reducing the unsustainable debt.
"Obama’s promise of a balanced, long-term combination of spending cuts and tax increases is the correct one.
"He will have to bring his own party along on entitlement reform, and persuade a dug-in Republican Party of the need for increased tax revenue not based on the wishful assumption of faster economic growth."
The influential daily, which had endorsed Obama's candidature, said there were other important pieces of unfinished business too, both in America and abroad.
"At the top of the list are comprehensive immigration reform, an enterprise that Republicans would be wise to join if they hope not to be made obsolete by changing demography, and climate change...
"Overseas, the Iranian nuclear programme will pose a fateful challenge, possibly within months.
"Obama will have to ensure that gains in Afghanistan and Iraq are not erased in the aftermath of US troop withdrawals.
"His dithering in Syria as 30,000 civilians have been massacred is a particular blot on his first-term record, one for which he could begin to make amends in the second."
The Post said the American economy was recovering, but far more slowly than anyone imagined in 2008. The re-elected president, it said, needed to get to work quickly.
"After the briefest of celebrations, the president will have to pivot to the looming fiscal cliff of scheduled tax hikes and spending cuts, lest the country veer back into recession.
"Still, the prospect of four more years offers Obama a chance to conserve the accomplishments of his first term and to complete its unfinished work."
"But the real measure of Obama's success, and the ultimate assessment of his presidential tenure, will be in whether, in a second term, he can fulfil some of the promise that made Americans so excited about his candidacy four years ago.
"Will an Obama second term allow him to transcend the ideological divides that he vowed to bridge but instead found so daunting?
"That is a tough order in a partisan age and with a divided, gridlocked Congress; there is no indication that the intransigence Obama encountered from the opposition party will diminish.
"But Obama has had four years of seasoning; one question is whether he can demonstrate the political canniness and legislative finesse that too often eluded him during the first term."