Solution 1:

Literal use of "last days of Rome" appears at least as early as this instance from "'Our Best Society'," in the Erie [Pennsylvania] Observer (April 30, 1853):

There is a picture in the Luxembourg gallery at Paris, "Decadence of the Romans," which made the fame and fortune of Couture, the painter. It represents an orgie in the court of a temple, during the last days of Rome. A swarm of revellers occupy the middle of the picture, wreathed in elaborate intricacy of luxurious posture, men and women intermingled ; their faces, in in which the old Roman fire scarcely flickers, brutalized with excess of every kind ; their heads of disheavelled hair bound with coronals of leaves, while from goblets of antique grace, they drain the fiery torrent which is destroying them.

From "Speculation! Speculation!!" originally from the Florida Union, reprinted in the [Bloomsburg, Pennsylvania] Columbia Democrat and Star of the North (November 28, 1866):

It cannot be but the Northern mind is astute enough to perceive that such a course [impeachment of President Johnson and reduction of the eleven states of the former Confederacy to federally administered territories] would terminate the career of the United Sates government, as the custodian of the world's l[i]berty ; and that from such violence of the genius of the Constitution, it would be no longer a life-giving power to liberty, but a m[a]sked battery, from behind which State's Rights and personal liberty would be effectually demolished ; for faction would rise up against faction, as in the last days of Rome when her too oft violated Constitution had lost its virtue to preserve Roman liberty, till this once favored and proud, "Land of the free, and home of the brave," would become Mexicanized—a by-word, and a hissing scorn to surviving nations.

And from "Life Worth Living," in the Pittsburgh [Pennsylvania] Dispatch (April 13, 1889):

"Life Worth Living" was the subject of a lecture by the Rev. W.R. Mackay, at St. Peter's Episcopal Church last night, in which he dealt very severely with the pessimist, as an introduction. Then he explained how a man's life depends greatly upon his physical condition. He said that pessimists were apt to take a rain-and-mud view of everything; that they produced hopelessness and despair, the source of suicide.

"Never since the last days of Rome," he said, "did we have so many suicides as at this period."

As the great causes of this, he cited materialism and infidelity as the characteristics of this, our scientific age.

In all of these examples we see "the last days of Rome" being used as a kind of archetype or standard of a period of national political and cultural decline toward extinction. The example of Rome's decline from early greatness—in particular during its era as a republic—and the ultimate destruction of the Roman empire by barbarian hordes has been of special interest to Europeans and North Americans for hundreds of years, especially when those countries began to develop empires of their own and to dominate the political world.

The subject has attracted renowned political thinkers such as Charles (Baron) Montesquieu (Reflections on the Causes of the Rise and Fall of the Roman Empire [1734]) and Edward Gibbon (The History of the Decline and Fall of the Roman Empire [1776–1789]). However, the exact wording expression "the last days of Rome" probably owes at least some debt to the popularity of Edward Bulwer Lytton's novel, The Last Days of Pompeii (1834), which tells a story in a setting leading up to the destruction of Pompeii as a result of the 79 AD eruption of Mount Vesuvius.

In modern usage, "last days of Rome" carries the same notion of cavorting through the end days of personal dissolution and cultural disintegration that it did in the Erie Observer 169 years ago. For example, from David Sheppard, On Some Faraway Beach: The Life and Times of Brian Eno (2009):

Similarly, tribalism and cultural turf wars defined late '70s New York City night life. This was a world epitomized on the one hand by cocaine-encrusted Midtown disco citadel Studio 54, which — in those pre-AIDS days of sexual abandon — had turned nightclubbing into a thing of last-days-of-Rome debauchery, and, on the other by the dingy, sticky-floored downtown rock dives where the music and decadence was sharper-edged, at least for those not having their angles bevelled by increasingly widespread heroin use.