The parish records which are available online for the Catholic parish of Kilbehenny go back to 1824 for baptisms and 1825 for marriages, so you might be able to find Michael's baptism recorded there, and then the younger siblings. Unfortunately, the marriage record might not be there. However, ages and years of birth (or marriage) as recited in later census records (in Ireland or the US) were often estimates, since most people did not know (and were not very concerned with knowing) exactly when they were born, so I recommend treating the dates you have as rough estimates. The records can be accessed at this link:
https://registers.nli.ie/parishes/0269
At times, if the local parish did not yet have a church building (as the Catholic infrastructure was being revived), people might be married or baptized in a neighboring parish which did have a church (I've seen this in my own family), so you could also check parishes near Kilbehenny. Kilbehenny is in the southeastern-most part of County Limerick, where Limerick, Cork, and Tipperary all come together. The parish to the south, Mitchelstown (in Cork) has records going back to 1792, so you might find something there for your family. You can access those records using the interactive map at the link given above, by clicking where it shows Mitchelstown. The records for Templetenny, to the east of Kilbehenny, don't go back quite that far, but may go back far enough for you to find a marriage record.
Unfortunately, parish marriage records usually do not show the names of the spouses' parents. However, you might be able to find Patrick's father in Griffith's Valuation (mid-1800's) or in the earlier tithe applotment books (1830's). If you find a relevant baptismal or marriage record, it may give the name of the townland where the family lived, and you could then check the other records for Walshes living in that townland. If, for example, you were to find just one Walsh in that townland (in the mid-1800's), then he could well be your Patrick's father (although it's also possible that he could be an older brother who had inherited the tenancy by then). At this link, you can see more info about the civil parish of Kilbehenny, with a list of links to info about the townlands in the parish (at those further links are yet further links to records from Griffith's Valuation):
https://www.townlands.ie/limerick/kilbeheny2/
The current Catholic parishes and the (older, administrative) civil parishes do not always have the same boundaries, or even the same names, but in the case of Kilbehenny the civil and Catholic parishes appear to be pretty much identical, so that removes one concern that sometimes creates confusion.