that involved a protagonist who was the only young person in a society
of decadent immortals
So at this point you'd be describing Against the Fall of Night, or as Gary posted, the expanded version, The City and the Stars. It is precisely analogous up to this point.
The story centers on Alvin, the first boy born in thousands of years. He lives in the last remaining human city, Diaspar, where the few remaining people do little or nothing. Their technology is so fantastic that they live forever, and huge underground computers do everything for them. Alvin, 17 years old at this point, is bored to tears living among the ancient. He goes off exploring, which starts off a series of rather unexpected events.
They lived in a floating city (or series of floating buildings), and
flying suits were ubiquitous. (One detail that stuck with me is that
everyone had pocket telephones, so at least one prediction came true!)
Unless The City and the Stars is radically different than Against the Fall of Night, which is what I read as a kid, this is where any similarity ends. I remember Fall fairly well, and none of these are part of the story.
Diaspar is definitely on the ground, and while they have tremendous technology, flying suits and cell phones were not part of it. The city was huge, miles high, and transport was by a sort of moving sidewalk that could also go straight up the buildings.
He eventually meets up with a woman with the improbable name of
"Synthetica" and discovers a society (with relatively primitive
technology) who lives underground in a fantastically-carved cavern.
This part is the same as Against the Fall of Night again. In that story, he manages to find a way to leave the city in a forgotten underground train system. The system map has lights to show potential destinations, and only Diaspar and one other light, for Lys, remain lit.
He travels to Lys and finds a thriving society of people who live in a garden-like paradise, not an enclosed city. Except for the underground part, once again we're on the same track. Here he meets Shalmirane, who is one of Lys' "rulers", and she introduces him to her son. The two go exploring and even more unexpected events occur.
So... either you read The City and the Stars and are mixing up some details with another story, or you read what appears to be a complete rip-off of that same work! The book was republished many times, the version I had I got new in the late 1970s or early 1980s. It showed the city on the front, and it took me years to realize it was in the shape of a face.