The Empire Was Intrinsically Unstable
If you stop to think about it, Julius' conquest was equivalent to Patton being sent to Africa, and coming back at the end of the war to invade.
It speaks volumes to the social rot that was already present in Rome at the conception of the Empire that this army of "sons of Rome" didn't care about marching on their own homeland, which was commented on by numerous people throughout the pre- and post- Empire
What Was the Rot?
The Roman sphere consisted of a broad demographic of non-Roman Italians, who were non-citizens. This led to the Social War, where these people took up arms to take citizenship by force.
Soldiers were paid in tips (basically whatever they were able to seize during warfare). Although, on paper, only citizens could join the legions -- because of how much the situation sucked, exceptions were commonplace. In an effort to keep the army solvent, high commander Marius allowed individual generals to take over the chore of paying their soldiers.
The government was beyond bankrupt, paying soldiers with promises of land that the government did not own (and which Augustus later simply seized from citizens).
What Did This Do?
By partnering with a wealthy banker and an established military leader, Julius Caesar was able to (on paper) make himself the richest person in the world (almost all of it was debt). By using his very real army to seize the government, he was able to seize the things he wanted.
Augustus made some almost prescient decisions, staying ahead of the political game, but keeping control of all that wealth, and the private army. Wisely, he exerted most of this control subtly, through ambitious intermediaries.
Tiberius never wanted the job, and basically retired after only 14 years on it. He allowed the social, financial, and military monopoly that his step-grandfather built, and his step-father maintained to erode; leading to a series of bloody power grabs : Caligula holding power for only 3 years. His two successors only holding power for a decade each. Then 4 Emperors in the span of two years.
After Tiberius eroded this power monopoly, there was never again an Emperor serving for Augustus' 40 years (although a few, like Constantine, came close at 30 years).
How to Make the Empire Last
Tiberius needs to care about the job. Augustus was a brilliant man, outsmarting many savvy politicians who were decades more experienced than himself.
I'd like to assume, based on this, that Augustus picked for the job of future Emperor someone who had the right skills and temperment. The only thing missing is that Tiberius never bought into the ideal of Empire.
So, imagining a Tiberius who cared :
- He would have adopted, like Augustus wanted, the role of "exceedingly rich and powerful citizen-servant of Rome", and would have used much of his wealth and power charitably to build and maintain infrastructure.
- He would have seen the growing tension between Christians and Romans and took some formal position
- He would have continued Augustus' practice of encouraging the ambitions of the political class, manipulating the Senate behind the scenes
- He would have swiftly picked and trained a successor who was not necessarily related to him, and worked tirelessly to - by increments - transfer all of the wealth, titles, and command legally to the new heir.
An invested Tiberius, when he died, would have passed along to Roman citizens 62 years of stable rule (40 years of Augustus', plus his own 22). Two generations would have grown up not knowing the instability and chaos of pre-Empire Rome.
Augustus had dealt with an Empire that didn't pay it's soldiers, and couldn't pay it's debts. Tiberius' reign could have put to rest the lost love for Rome brought about by the Roman governments' misbehavior for the last several decades.
Now, if Patton went to Africa at the head of the army, they would not have his back if he chose to invade Washington.
This stability could have been lost by successors, just as the Republic had originally lost it.
Growing Into Space Rome
The Dark Ages, we've started to discover, really weren't that dark. Rome collapsed, but Roman cultural influence continued in the former Roman provinces. Global trade continued, even though it was not supervised by Roman authorities.
A stable Empire would have remained "Julian"
The problem Rome always had, since before Julius, was that the government did not bring in enough money to cover the cost of the services it supposedly provided. Rome was always broke.
A stable succession of "super wealthy and powerful 1st citizen-servants" would need to have presciently solved this problem.
I can imagine inventing banking before the Templars (1100 AD) would have been a good start. Other investments that could have added value to the Roman bottom-line would have been an early invention of Hospitals (340 AD), sewage systems (Paris - 1370 AD), dedicated police and investigative forces (Paris - 1667 AD), the specialization of psychiatry as a discipline (1900s AD, but ideas were around since BC).
The Antikythera mechanism dates to 150 BC, indicating that both sophisticated computing and automation pre-dated the Empire. Without so much political churn, these ideas could have been re-discovered way ahead of Charles Babbage and Ada Lovelace in the 1830s.
Provided the Emperor could continue to figure out how to provide enough wealth to keep everything stable (and grow a technological society).
The Library of Alexandria, containing all of this information and more, may not have been burned in this more politically and economically stable climate. Using the stars for navigation also pre-existed Rome, so the idea of a spherical Earth was already around, even if nobody applied math to it.
There's no particular brilliance in using alcohol as rocket fuel. If the Empire had trade routes along the silk road, they would have picked up the idea for rockets (222 AD).
With that, everything would be in place for a stable Rome to be shooting for the stars in the 200 ADs. There would have to be a compelling reason for the "1st citizen-servant" to pay for the development, but there are lots of possible reasons.