The question of whether the papar of Ari were real people is actually still a fairly hot topic in Norse studies! There is some evidence to indicate that they were real. However, there are also very good reasons to doubt Ari's account.

First, the evidence for: there are two main pieces of evidence to suggest Irish presence in Iceland beyond Ari. The first is place-name data: there are a lot of places around the island that have a placename "pap-" (e.g. Papey, an island off the coast of the Eastfjords). This translates to "of the papar" and Icelandic place-names tend to have a lot of stability over time. This then, at least sometimes, suggests that there are very old stories about the papar associated with these places, and it becomes at least possible they were there. The other main piece are the caves of Seljaland. Kristjan Ahronsson has been mapping out caves discovered on a private farm in Seljaland. These caves are 1) hand-excavated and 2) filled top to bottom with carved crosses. He argues that the oldest of these carved crosses are stylistically identical to crosses carved into stone in the Hebrides around 800, and we should therefore interpret this site as roughly contemporary to that. I've been to that cave and spoken with him about it, and it is a genuinely fascinating site. But, the dating evidence is a little weak. It is finally worth noting that by no means are they the only Papar-site with carved crosses, but it is the most impressive.

There are plenty of reasons to doubt that the papar are real, though. First off, place names aren't universally reliable - some of the papar placenames in the Eastfjords appear to only date to the 16th century, and appear to be due to calling irish merchants visiting the island "papists". Additionally, providing the chronology of crosses is incredibly tricky, and in this case, getting the date even a century off would require a complete reinterpretation of the site. Almost from the beginning of settlement, there was a Christian (and often specifically Irish) presence among both free and enslaved people, and, in fact, other caves in the Seljaland area and the Westman Islands were probably used by enslaved people who revolted against their owner's abuse. It is well within the realm of possibility that these people are the ones who hollowed out the rocks and carved crosses while waiting out the brutal storms that are fairly frequent in the area (I am speaking from personal experience waiting in that cave, btw - really communicates the atmosphere of just how deliberately miserable the monk-hermit life was.)

I'll conclude this answer with a note on why Ari might have wanted to make up the papar. It's notable that in Islendingabok he says "they existed, Norse people arrived, they peacefully and mysteriously vanished." As if they were recluses who just didn't like the noisy new neighbors. There's this very mysterious and ethereal nature to them, but the thing Ari says they definitively left behind "books, bells, and staffs" - i.e. sacred objects. Ari is very worried about Iceland's status on the very periphery of the Latin Christian world (as, indeed, many monastic saga authors appear to be), and about the pagan past of the island. So, the papar as Ari talks about them have been interpreted as a bit of religious cheating - the original inhabitants of the island were Christian, therefore Iceland is properly Christian, therefore we're well-integrated into the bounds of the Christian world and not where all the corrupt "monstrous" peoples live (drawing on people like Isidore, whom Isleifr Gislason, the first bishop of Skalholt and one of Ari's named sources, would have likely read during his education in the Holy Roman Empire.) Given that Ari is the attributed author of both Islendingabok and the first version of Landnamabok, and all other textual sources that mention the papar seem to derive part of their information from those two, it is possible (though I think unlikely) that they are a pure fabrication.

So... in short, we don't know. There has been a lot of scholarship on both sides of the issue, and the evidence on their existence is pretty evenly balanced either way, so i don't see the field reaching any sort of consensus any time soon.

Insular Christianity as was practiced by many Irish monks was in many ways similar to the Christianity practiced on the mainland, but seemingly very, well- Isolated, by its own unique facets.

We don't really have concrete numbers on the amount at any given times, but littered throughout contemporary Christian chronicles and as you mentioned, Norse sources, we see mention of these earlier Monastic communities. These are often of dubious quality as they were often written by new figures in an old and often dangerous land, which was then followed by years of mythologizing. Even so, there does seem to be consistent theme of small, insular communities throughout the Irish Sea, the North Sea Islands, and possibly even Iceland as you mention.

As a side note, I recently saw a video wherein researchers claimed there was record of human habitation on the Azores Islands as early as the 950's AD; and while it could be a great variety of peoples, they were gone as a community by the time of the Portuguese, leading very nicely to the theory that they were a group of Irish Monks-people who have no way of sustaining a population. There was very little material culture, and no evidence of violence, only trace remnants of the same organic compounds as was produced by Northern European Mice, Sheep, and Cows.

Leading to my next point- The culture. Life on a monastery was not designed to be something marveled over, it was purposefully designed to bring about a simple life where one would not worry about the material frets of life. This was especially true of many Irish Monasteries, where Papal interference was perhaps lowest for a large period of time. These communities would have been even more isolated and insular had they been located on a place such as Iceland, wherein the whole point would have been to escape the struggles of the lay entirely. These communities would have been much more in line with what we today would imagine as a Buddhist monastery; Nearly self-sufficient, and only accepting those who were inspired by their way of life- not actively proselytizing in an attempt to expand.

To answer your question then- Not much; as we would understand it. Depending on the size of the monastery in question, they may have had livestock which needed tending, gardens, and regular repair work for their simple dwellings. But at the end of the day, these men would have come to a monastery for the life of a hermit- or as close to it as they wished; men who would regularly fast and go without any of the conveniences you and I cannot go a day without. There were larger associations sure- as many of the larger monasteries near urban settlements in Ireland proper had whole scriptoriums for the replication of literally golden Bibles, and actively worked with their lay communities. But the difference between the two are night and day; they may have both come from the traditions of Saint Anthony, but as it is today, peoples interpretations of what they do for faith are often paradoxically opposed, even when united.

Hope that helps, tell me if there is anything that you would like elaborated!