Ho come most of Ireland was christian by the 6th century despite little to none Roman contact, and Irish missionairies had to christianize other parts of Europe (Great-Britain, Low countries, Germania, Gaul) that had more contact with Roman culture and thus christianity?
First, while, yes, the majority of Ireland was converted by the end of the fifth century, early Irish Christianity was super weird. Very little earlier mythology seems to have been completely abandoned, although the Irish did try to claim some romanitas by insisting that the founder of the "Irish race" was a Roman by the name of Mil, that is, Militus, that is, "Soldier". Irish saints are particularly fantastic, and behave much more like demigods than the models of sanctity you find on the continent. Continental hagiography (stories of saints) was very concerned with holiness and piety. Irish hagiography consists of a whole lot of cursing and anathematizing with a couple miracles thrown in for good measure. It is extremely unwise to piss off an Irish saint. As one of my compatriots said when we were discussing this post, "Most of St. Patrick's vita (life story) is him killing wizards with magic Jesus power."
Second, Irish missionaries didn't Christianize Europe, they monasticized it. Europe, at least within the former boundaries of the Roman Empire, was certainly Christian, and even England and Scotland were reconverted after an Anglo-Saxon interlude starting with the mission of St. Augustine of Canterbury sent by Pope Gregory I in the first decade of the 600s. Wales is a slightly different case due to its proximity to Ireland and the frequent travel between the two areas. In fact, when the Irish monks began expanding outward, first in Wales in the sixth century and then into England in the seventh, they were found to have particular doctrinal deviations from Roman practice, specifically with the proper way to calculate the date for Easter.
The real expansion of Irish monasticism came with the foundations of the abbeys of Luxeuil (ca. 590s) and Corbie (ca. 660) in modern-day France. These served as the foundation for the revitalization of western learning after the collapse of the western Roman Empire. Almost every known script in Europe during the seventh and eighth centuries was derived from those used in the scriptoria of Luxeuil, Corbie, and Benevento, all of which were Irish foundations, and whose scripts were themselves of Insular extraction.