Irishkiwi,
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find a statutory birth, certificate for James. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
You haven’t said what denomination James was but if Roman Catholic, the records for Pomeroy parish start in 1837 (with some gaps). So you might just scrape in and you might find some siblings born after him. The surname was often spelled Kane so bear that in mind too. Spelling of names varied all the time in Ireland and folk didn’t always use the same one. Likewise they used and dropped the O’ prefix on a whim, so it could be O’Kane or O’Kain in some records.
I had a quick glance at the Pomeroy records and I do see a baptism on 9th Feb 1838 of a Mary Kaine to Laurence Kaine and Isabella Grant. Sponsors were Daniel McGladrigan and Mary Gate (?). The couple’s townland (address) was Pomeroy. So that implies they lived in the village of Pomeroy, not just the parish. You might want to see if there are nay other baptisms to that couple.
I looked in the statutory records for a death for Laurence, 1864 onwards in the Cookstown area (which includes Pomeroy). I did not find one. So he may have died before 1864. There was an Isabella Kane who died 19.1.1884 aged 84, widow of a bootmaker. Daughter Ann Jane Kane was the informant.
https://churchrecords.irishgenealogy.ie/churchrecords/
There was another death in 1877, aged 60, which isn’t free to view yet, but if you want to view it you can do so for £2.50 on the GRONI website.