Patrick Carroll is a very common name in Ireland.There were 854 people of that name in the 1901 Irish census. 25 people of that name died in 1885, 2 in Offaly but neither was born anywhere around 1828. There were others who died elsewhere in Ireland who were born around 1828, but there would be no immediately obvious way of knowing which, if any, was the right one.
You’ll probably need to get more information from US sources, to try and narrow the search. EG naturalisation documents, or US censuses.