Octavian's early career was not only anomalous, it was outright illegal under normal circumstances. First of all, armies could not legally be raised and led by private citizens, only by elected magistrates and promagistrates holding imperium. Moreover, Octavian received the consulship on August 19, 43 at the age of 20. The minimum age for the consulship, as established by the lex Vilia in 180, was 43. Furthermore, Octavian had not held the quaestorship, formally a requirement for entry into senatorial magistracies since Sulla, and was not even of age to offer his candidacy, as the minimum age for the quaestorship was 31. Octavian's meteoric early career was made possible, if only barely acceptable to a relatively few, by special provisions passed in response to the state of intense political feuding and near-anarchy that Caesar's murder left the Republican state. Octavian arrived in Italy by the fall of 44 a complete newcomer to politics. He had never held any magistracy and only with great difficulty--and the aid of Cicero, who was strongly opposed to Antony--managed to secure the fortune left him in Caesar's will. As Cicero delivered the Philippics condemning Antony and eventually driving him from Italy entirely, Octavian was gathering a private army of his own accord and from his own (inherited) fortune, consisting mainly of Caesarian veterans and serving Caesarian legions which defected from Antony. This army was entirely illegal, as Octavian had no legal status allowing him to maintain an army--he was not even a senator yet, much less a praetor or consul. But with Antony gathering his forces in Cisalpine Gaul, Cicero and his supporters saw Octavian as something of a necessary evil, an ally on which they had to rely in order to provide themselves with an army capable of combating Antony's. Octavian was thus, on Cicero's recommendation, granted an extraordinary propraetorian magistracy and furthermore afforded a senatorial seat, a consular veto, and the right to stand for every magistracy ten years early (subsequently violated in his first consulship). By sheer coincidence, both consuls were killed at Mutina, and Octavian was left holding a bag of two consular armies and some assorted private troops, a force with which he was able to force a consular election in August which elected him--fully 23 years too early--for the consulship, alongside Q. Pedius.

This was not normal. Classical Roman ideology held that although boys came of age around 15 or 16 they were not yet fully men until they had passed 30 or so. The adulescens was not, as in modern English, an adolescent, but he was still a man too young to be entrusted with independent action. In practice, at least for the nobiles, this generally meant that until the age of 30 or even older a young man was still considered not fully equal to his elders and certainly not fit for great responsibility. This is the time of life during which it was appropriate for a young aristocrat to enter into a military tribunate and learn the ropes of leadership and politics from his fully senatorial superiors. This has nothing to do with life expectancy. Average life expectancy in antiquity was reduced by premature death, either as an infant or due to disease or warfare at a young age. Provided one survived to adulthood the chances of reaching an advanced age were pretty good--Augustus himself lived to the age of 76. Instead, adulescentes were simply not considered mature or experienced enough to be entrusted with responsibilities. And indeed, as I have already pointed out, legal prohibitions prevented the entry of an adulescens into the cursus honorum--and thus, the senate, following his term as quaestor--before the appointed age for precisely this purpose. Octavian's case was not wholly unusual, however, if totally unprecedented in the speed with which Octavian sought the consulship. The lex Vilia had been passed in the wake of the Second Punic War, which saw a number of politicians rise exceptionally quickly through the political ranks. Scipio, for example, was elected to the consulship in 205, when by the lex Vilia twenty years later he would have been more than ten years early, and five years before in 210 had been granted proconsular imperium despite never having held praetorian or consular rank. Closer to Octavian's time (and, more importantly, following the establishment of minimum ages and Sulla's provision for mandatory quaestorships), Pompey held the consulship without having ever held any previous office or even having a senatorial seat. Pompey's career was arguably even more radical than Octavian's. Pompey had raised three Sullan legions from his own clients and dependents in 83, despite being a private citizen--this could only be allowed by the great instability and disregard for legal guidelines caused by the civil war. Pompey was granted his first triumph by Sulla in 79 (at 27 years old), despite being a private citizen, not being a senator, and never having held any elected office--technically speaking not altogether legal. Pompey's first elected office was the consulship, which he held in 70 at the age of 36. Both facts should have disqualified him for the consulship, but Pompey had enough influence at that point that the rules were bent--a not altogether unheard-of feature of the Republican state! But these cases were all anomalies, and the Republican cursus honorum was not supposed to function like this. Only crises, civil wars, and other instabilities granted these individuals the ability to enter into high-level political careers earlier than they were traditionally--indeed, legally--allowed.

OP, I just wanted to comment on this part:

Right off the bat, it seems like shorter life spans would significantly alter what it means to be "young".

Life expectancy is an average. Treat it like that. It did not mean that people grew older faster and that by age, say, 40, you were a senior. It means there were many more reasons to die young.

Infant mortality was especially high, bringing the average down. Then, when you were adult, there were more reasons to die young than today, such as lack of nutrition, illnesses, warfare etc.

But in the case of a rich, urban roman such as Gaius Octavius, if he made it out of infant years he could expect to live quite old... Gaius Octavius himself died at the ripe age of 75, and his successor Tiberius lived to be 77.