There were two primary mechanisms by which inflation happened in the ancient world. Let's use Rome here for examples.

The first was by shortages of a product, leading to price jumps. This was particularly common during the early Republican period, but also during later eras, and primarily focused around inelastic products like grain. The way that they 'solved' this was through price controls and the grain dole. The former, like most price controls, was not a particularly good solution, while the later was simply imposing state will on a sector by leveraging the massive treasury and of course military might of the government. The grain dole is one of the primary reasons a city like Rome could have a million people in it at the height of the Empire.

The second mechanism is by currency devaluation. In the modern day, this is achieved by simply printing more money. In an era of precious metal based currency, it is achieved by debasing (that is, reducing the precious metal value) of the coins. The primary victim of this was the denarius, a coin made of silver. This coin dropped from at least 90% pure silver to less than 10% by the late imperial period. Needless to say, when you still have coins that are 90% pure silver circulating alongside ones that are 10%, and you attempt to call them the same value, trust in the currency, and inflation of prices will result (or, more accurately, deflation in the value of the later coins). The choice of the denarius is pretty obvious, for the record, because it was the way Legionaries were paid, and that was the single largest outlay of the Imperial treasury, by a wide margin. In contrast, gold coins remained relatively pure, and being largely the per-vue of the richest men (being far too valuable for most day to day transactions) were not debased. Several emperors attempted to mitigate this to varying degrees of success, but putting the genie back in the bottle, but once confusion is introduced into a precious metals based currency system, it's very difficult to restore. What seems like an easy solution (debase the currency so my 1,000,000 denarii is now 1,500,000 denarii) for emperors leaves their successors with growing problems.

Inflation didn't exist in ancient Rome or Egypt, because there was no paper money. Currency was based on gold or silver, and you can't simply print more of these. One rare case where there was inflation of gold currency happened in 1324 AD when Musa I. of Mali arrived in Egypt during his pilgrimage to Mecca, he spent so much gold in Egypt that gold vas devalued in the region for the next decade and the prices of all goods and wares were inflated. However, that was some 1000 years after the end of Antiquity.