Currently there are several websites from which to begin your search for newspapers in Ireland and Northern Ireland. None are currently complete, but together you may find what you are looking for.
One up and coming website is “Irish Newspaper Archives” The INA currently has over 2 million pages from 23 titles, making it a website to constantly refer back to see what has been added. They have microfilm for 30 additional titles to be added over the next year.
Another website is the British Newspaper Archive: www.britishnewspaperarchive.co.uk which has newspapers for both Ireland and Northern Ireland. This website is part of the British Library.
A third major website is the Newspaper Archive: http://access.newspaperarchive.com This database has a large United States presence, but also has a growing worldwide section. They currently have newspapers for both Ireland and Northern Ireland.
For any growing database, sometimes it helps to know what you are looking for. In this case I would refer you to the “Newspaper Database” on the National Library of Ireland website: www.nli.ie/en/catalogues-and-databases-printed-newspapers.aspx This database is taken from the NEWSPLAN Project which makes an effort to microfilm and preserve newspapers in Ireland and the United Kingdom. The resulting database can be viewed by locality, which will then list the newspapers published there, and what years are covered. It can also be viewed by county. From this inventory, you will then know if the newspaper you want has been placed online or if you still need to look at it on microfilm at the National Library of Ireland or another repository.
All of these re growing websites with millions of new pages added on a regular bases. Be aware that not every newspaper is currently available, and you will need to check back to each website to see if what you are looking for has been added.