What would be the chances of the Romans being able to figure out cast iron?
There is no
actual obstacle. The Romans were using bloomeries, as the Chinese had done prior to their invention of the blast furnace in the fifth century BC. Europeans also invented the blast furnace... somewhere between 1100 and 1200 AD. A relative dearth of ironware dating from before the fifth century BC in China has led some to theorise that the Chinese had a relatively short "bloomery phase", and invented the blast furnace rather early on. Europeans, on the other hand, used bloomeries for much longer, and invented the blast furnace much later.
So this is a big random factor. My objection would be that if you give the Romans (or anyone) a lot of "luck" in having them just invent stuff as early as it's reasonably possible, you soon get into wank territory. On the other hand, if a certain (pre-)industrialisation is already underway, it becomes more likely that people start looking into new methods and techniques. So ultimately I'd say that having a blast furnace (and thus the possibility of cast iron) invented relatively soon after the POD (as in: within three or four centuries of it) smacks of wankage, while having it emerge in Europe a few hundred years before OTL's date of the 12th century is very reasonable.
Another option, of course, is that earlier developments in the ATL lead to increased contacts with other cultures, which would realistically allow the Romans to copy certain techniques and inventions. China is the obvious candidate, but very far away. I would note that the Haya people, living on the southwestern shores of Lake Victoria, used blast furnaces 2000 years ago, in Roman times. And Roman coins have been found on the eastern shores of Africa (in Kenya and Tanzania), where Roman traders apparently followed the example of the Greeks in trading with the "Azanians" (East Africans). Ancient expeditions along the Nile invariably got no further than the
Sudd, but in an ATL where (pre-)industrialisation has already had the right effects, one might realistically have the Romans set up trade posts in East Africa, launch a few expeditions inland, come across the Haya people, and then figure out that those blast furnaces these guys are using are
really awesome.
It wouldn't be the first time Romans came across "barabarians" using some cool tech, and just copied it. They had a habit of doing so in OTL. "If it is of use to Rome..."