Surveys suggest that support for Mr Abe's government has slumped to little over 30%.
Mr Abe's current emphasis on stamping out amakudari suggests a tin ear for the public mood.
Opinion polls gave a swift verdict: Mr Abe's popularity, which had plumbed abysmal depths, shot up.
After the meeting, Mr Abe and Mr Obama dined with US Vice-President Joe Biden.
In December, Mr Abe and Mr Putin agreed to restart talks on signing a peace treaty.
Mr Abe insists that revived growth and lower government spending will plug the gap.
Mr Abe threatened to revoke the central bank's independence, if necessary, to achieve all this.
Under pressure from Mr Abe, Japan's central bank doubled its inflation target to 2% earlier this year.
Mr Abe offered a "masakaki" decorated wooden stick, inscribed with his name and title, Kyodo News agency said.
The second consequence of Mr Abe's policies is a dramatic fall in the value of the Japanese yen.
Since Mr Abe's election, Japan's stock market has taken off, rising by 30% in the last three months.
Mr Abe has dropped his constant theme of national pride and now listens, he says, to economic concerns.
That Mr Abe can even think of carrying on gives an idea of the confusion in his party.
So Mr Abe too needs to show that the region's two most powerful countries can rub along more peaceably.
Moreover, Mr Abe's move to tap the party strongmen should be seen as just another sign of his fragility.
Mr Abe probably calculates that by spreading favours he can stop a party in disarray from turning on him.
Mr Abe has already served as Japanese prime minister between 2006 and 2007.
On top of that, Mr Abe's government, though short-lived, was dogged by a seemingly endless series of financial scandals.
Mr Abe said the process of dismantling the plant, which was decommissioned after the disaster, was an "unprecedented challenge".
BBC: Japan PM Abe praises Fukushima nuclear work during visit
And that is something Mr Abe's government will have to get used to.
However, since Mr Abe promised to stimulate the economy by spending, the yen has fallen almost 6% since November.
Mr Abe has called upon the bank to raise its inflation target to 2% - double its current target.
The reign of Mr Abe's starry predecessor, Junichiro Koizumi, helped disguise this failure.
However, Mr Abe pledged to address those issues and said that the government would ensure that it safeguarded Japan's interests.
To deal with the hideous cost of farm subsidies, Mr Abe has appointed an affable farmer's friend as agriculture minister.
Mr Abe is serving as Japan's top leader for a second time, after a brief period in power in 2006-7.
Even his political godfather, Yoshiro Mori, a former prime minister, seemed at the weekend to give up on Mr Abe.
Mr Abe "is not fully aware of what he has said and what the implications could be", Mr Sakurai added.
Mr Abe has backed the central bank's ultra easy monetary policy, which calls for pumping more money into the Japanese system.
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