If Caesar arrives in Gallia in 500AD with the army he had at Alesia, he would find the region split in 3 parts (pun intended). The Franks in the north, the Visigoth in the south and the Burgundians in the southeastern corner.

He could easily beat all three one by one. They do not love each other at all.

Franks and Visigoths were almost constantly at war. During the largest battle of these conflicts in 507, the franco-burgundian coalition had around 15,000 men, and the Visigoths close to that number. Caesar had 60,000 veterans and auxilia at Alesia and more soldiers in various depots and garrisons around Gaul. He could likely conquer Gaul alone in a few seasons: the Gaul of 500 was still deeply Romanized, most of its population was Roman and it had Roman roads and cities, facilitating fast movement and supply. Clovis and his warbands won constantly, but they never met an army like Caesar’s veteran legions.

The next step would be to finish defeating the Visigoths and freeing Hispania, while consolidating its rule in Gallia. Even that step should be easy. Caesar knows Spain well, and again he would have a large superiority in the number of trained, disciplined and better equipped soldiers.

Italy, under the Ostrogoths and, formally still a part of the Roman Empire of Anastasius Augustus, would be harder to conquer: full of cities that need to be taken one by one, logistically difficult. Taking Ravenna without a fleet would be a nightmare for example.

Belisarius learned that the hard way himself. But he almost did it with an army that was, at best, just one quarter the size of what Caesar would have. Theodoric is a good leader and general, but Caesar is still superior.

Fortifications can delay Caesar’s conquest. There are more of them and often more complex and harder to take in this century than in the 1st century BCE. Still we all know classical Romans were not slouches when siege warfare was involved, so fortifications would not be enough to stop him.

The real problem is Africa. The province is critical to the functioning of the Empire. But it is hard to reach without a fleet. Fleets are expensive to build and to maintain. And there is no fleet in 500AD, in either Italy or Gaul that can move 20,000 soldiers together and that is the minimum needed to take the province I believe. .

A failed attack of Africa, by sea, in 468, almost bankrupted the Eastern Empire. Not sure where Caesar finds the ships and the funds he would need three decades later.

And this naval bottleneck is a good segue to Caesar’s main problem. A logistics problem, not a military technology one. Military tech of the 5th century is very similar to the one of Caesar’s age. Better under some aspects, but worse under others. Superior iron age military tech can be quickly copied and adopted by your enemy.

To maintain control of the army, Caesar needs money, food, supplies. Those would be much harder to come by.

A Roman army of the I century CE worked because it had huge resources to pull from. Caesar’s army was sustained by the combined resources of three heavily taxed provinces and the spoils of victory. Here the spoils would be limited and he would have no provinces to provide the funds.

This army can likely live initially by simply eliminating the German tribes living in the western provinces, take their weapons, work them as slaves to produce goods and pay its soldiers.

It must conquer lands quickly and secure a revenue base and food supply or risk starving, but what happens when there are no new areas to reconquer?

Would Hispania, Gallia, Italia, Africa (if you can take it) and Illyricum, with their 500 CE economy able to sustain an army of 60,000+ and more necessary to control the territory and defend the borders?

It might, but it is definitively not something I would be sure about. Some reforms, maybe giving land for service will be necessary.

Also what would the East do, knowing Caius Julius Caesar himself is knocking at their door?

They know history as well as we do, probably better, with documents we do not have and they might find the idea terrifying and decide to interfere.

Even more so given that this Caesarean army is a republican, pagan army. They are almost as alien to the mentality of a 5th century Roman as the Rome of the kings is to them.

Would the Christian populations of the provinces, who had trouble accepting the Arian Germanic tribes, accept a bunch of pagans as rulers?

How would Caesar legionaries live in a world where Romanity is widespread, but where their temples have become churches to a god they do not know? Where their Republicans ideals are polluted by hundreds of years of dominate and autharchy. Where even the language spoken every day is as different from theirs more than Shakespeare’s English is to us and wouls require efforts to understand (no to speak of the growing difference in regional accents).

Would the elites in the Roman world, oppose Caesar, just like they did in his times?

These issues might doom Caesar much more than any military opposition.

He might conquer the west back in 3–5 years. He did it already in the years between 48 and 45 BCE. But can he keep it? I have my doubts.