Hearing about the adulation behind the 1980s anime "Maison Ikkoku"（めぞん一刻）all these years, I think that the various opening and ending themes have probably added to the show's fame as well as getting fame themselves due to the popularity of "Maison Ikkoku". Of course, the show's got the moody opening theme "Suki sa"（好きさ）by Anzen Chitai（安全地帯）, and then the ending theme "Cinema"（シ・ネ・マ）by Picasso（ピカソ）.
According to the J-Wiki article on the song, Picasso's 6th single was "Begin The Night" released in September 1987. It also served as the final ending theme for "Maison Ikkoku", and unlike the atmospheric tango of "Cinema", "Begin The Night" was more of a conventional pop ballad, perhaps similar to the output of pop band H2O at the time.
Written by Etsuko Kisugi（来生えつこ）and composed/arranged by Picasso, "Begin The Night" does sound contemporary 80s but I can also hear a bit of a 1950s sound at points as if Richie and Mari Beth are holding hands on that porch swing. The slight echo in the song also gives it a haunting quality as if it was indeed the end of the story of the famous boarding house.