Un certain âge is a polite way of saying old, or at least not young. Certain here means a certain amount, as in not insignificant. It also adds, funny enough, a modicum of uncertainty. We’re not sure how old exactly this person is, but not young anyway.

Plus très jeune is to be taken as a one-word expression, if you will, not a grammatical construct. You can and should say Cette personne n’est plus très jeune. But you’ll say plus très jeune, pas très vieux, etc…

The important factor behind these expressions is the social unacceptability of referring to an old person as just being old, and therefore the search for roundabout ways of expressing the age of the person. This is already present in the original — ältere instead of alte – as well as in the English translations — older and elderly instead of old.

D’un certain âge and plus très jeune are two such expressions which rely on conveying meaning by way of a sous-entendu, i.e. a second, real meaning “underlying” the literal meaning.

• une femme d’un certain âge here means “a woman of some years”: meaning her age is not known, but it’s a somewhat non-negligible quantity. The underlying meaning is old, and is in the same register as English an older woman.

• plus très jeune is a litote which aims to emphasize the age of the person by negating its opposite. To me, this actually sounds harsher than older, though still not as harsh as vieille femme.

Note that the literal translation “une femme plus vieille”* does not have the same meaning. It implies a comparison to another (specific) person, which isn’t the case here.

Another term that could have been used here is une femme âgée (which means an elderly woman, not an aged woman).

Other options like “vieille” or “agée” are way too direct, and not polite towards the elderly woman.

1. “D’un certain âge” is an euphemism for “pretty old”. Wordreference gives “of a certain age” as a translation for exactly that. It’s a number we don’t exactly know but it’s not a small number.

2. We don’t need “ne” as there is no verb. If you say “une femme plus très jeune” you don’t use it, but you can change the sentence a little to fit it in: “une femme qui n’était plus très jeune…”
   I think you can see how “not so young” can express being old in a polite way.