Were there civilizations that had population decline problems similar to Japan's (due to ageing, fewer marriages, etc.) and what happened to them?

In the late Roman Republic and early Roman Empire they believed they had such a problem. Augustus enacted laws to encourage marriage as well as showed off people with many children as an example.

Some laws he abrogated, and he made some new ones; such as… that for the encouragement of marriage… The equestrian order clamoured loudly, at a spectacle in the theatre, for its total repeal; whereupon he sent for the children of Germanicus, and shewed them partly sitting upon his own lap, and partly on their father's; intimating by his looks and gestures, that they ought not to think it a grievance to follow the example of that young man.

Seutonius, Augustus, XXXIV

One way they encouraged marriage and the begetting of children in the upper classes was by favouring those with many children for high office or for plum assignments in rich provinces. Yet there were frequent dispensations, and prominent men were often given those offices even though by law they shouldn’t have them.

I claimed my rights under the lot and, when by virtue of having more children another proved to have the prior claim, I was… satisfied with that most splendid province which was left to me…

Fronto I, Loeb, p. 237

These laws probably had little effect, as Fronto’s case proves. But traditionally Rome had been very open to immigration, even opening an asylum in early Rome and sometimes forcing captured peoples to take citizenship. Later we have a speech from Claudius as he tried to convince the Senate to allow Gauls into their ranks (Lyon Tablet).

They freely gave citizenship to worthy foreigners and even in later years brought in whole tribes of wandering peoples within their borders. This helped provide manpower but it also caused problems, so I would say that the result was mixed, based upon the time period and the exact locale one considers. But overall, I would say that Rome's strategy of openness helped their population problem, at least until close to the end of the West.