I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church/civil/census/etc B/M/D records for MOST Irish counties, including Kilkenny. Records before 1863 are increasingly sparse - almost non-existant before 1800.
For a Nicholas Carroll and Bridget Martin marriage in 1830 +/- 4 yrs, Co. Kilkenny, I found 1 record: 1832. The parents' names are not shown on this record. A view of the record (at a small fee) MAY show a few more details.
If your DOB for Bridget is correct, she married VERY young, especially for Ireland. Due to the age disparity, Nicholas may have had a prior marriage. If so, I can't find it in transcripts of the surviving records.
For Bridget Martin B 1818 +/- 4 yrs, Co. Kilkenny, with the parents' names unknown, I found 1 record: 1818. The father's forename is Denis, mother's Ellen. Again, a view of the record MAY provide further details.
For Nicholas Carroll B 1805 ditto, I found 1 record: 1805. Parents' forenames for this record are Richard and Elizabeth. Again, a view . . . .
For their port of embarkation, they likely sailed/steamed from Cork City. Another favorite route was to sail to England and book passage in Liverpool. Also some ships departing from Liverpool took on passengers in Cork enroute to America.