This question touches on an issue which has caused a serious amount of real and digital ink to be spilled, occasionally in anger. And I hope some other commenters come to add to what I'm going to say here, because, as an archaeologist, I might leave out or oversimplify the linguistic parts of this conversation.

So, to get down to it, no written source prior to the modern period ever refers to the people of Ireland as Celts or Celtic. Medieval Irish sources do refer to Irish people using the words Gael and Gaelic (though those are the modern Irish post-reform spellings), so we know that Gael is a historically accurate word for the people who lived in Ireland.

Celts and Celticness get added to the mix due to the reliance of antiquarians and historians from roughly the 17th century onward on classical Greek and Roman texts to add names and 'flavour' to Northern European peoples who had little to no writing before the Medieval period. Added to the fact that people began to do comparative linguistics and realised that the Irish and Scots gaelic, Welsh, and Breton languages were all closely related, along with the assumption that Breton was the last surviving holdout of the language of the Gaulish people (it isn't, it was introduced around the 5th century AD from Britain), the idea took shape in the minds of scholars that the Irish/Scots, Welsh/pre anglo saxon Britons, and the Bretons were the last surviving descendants of the people who Classical authors called Celts or Gauls.

This dovetailed fairly well with the archaeological work being done in the 19th and early 20th century, which worked along what we call the 'cultural-historical' model - that is, that ethnicities in the past could be identified through their use of specific suites of artefacts and burial traditions, and that the appearance in the archaeological record of those things indicated the movement or spread of those peoples from one place to another. How this is of relevance to us is that by identifying archaeological material that was deposited immediately prior to the deposition of Roman archaeological material in places like France, these early archaeologists identified the bundle of artefact types that they labelled as being typical of the Celts. The most important part of this bundle was a specific art style mostly executed in objects that would have been used by the elite class of this society such as weaponry, feasting equipment, and personal ornamentation such as brooches. This art style was labelled 'La Tène', after an area of shallow water in Lake Neuchâtel in Switzerland where extremely large quantities of artefacts fom this period were sacrificed. From this evidence a grand theory emerged which viewed the celts as a central European people which spread from South Germany/East France/Switzerland from around 500BC, violently encountering the Classical world, and settling in Britain, Ireland, Iberia, the Balkans, and as far east as what is now Turkey.

To tie it all in to your question about Ireland, what we have by the end of the 20th century is the equation of a language family, an archaeological artefact category, and a category of people described by mediterranean authors in the classical period. It was 'known' that the people of Ireland must be Celts because their language was related to pre-germannic british languages and both these areas have archaeological artefacts that are also found on the continent in areas where the Greeks and Romans identify a group of people who they labelled as being 'Celts'. Ergo, Gaelic Irish culture must be a direct continuation of the Celtic people who, depending on your viewpoint at different times in the modern period, were the original inhabitants of Ireland, or had invaded in the Bronze or Iron Age. These ideas also tied in to theories of racial identity and national identity and was particularly important for national liberation movements in places like Ireland and Brittany.

However, by the 1990's some problems are arising in this idea. Chiefly the fact that there is no archaeological evidence for the mass movement of people from Britain or Continental Europe to Ireland. While the appearance of La Tène art in Ireland circa 300BC is one of the defining parts of the middle and later Iron Age they are mostly of a local variation (called insular la tene) that is more evidence of the adoption of this style by the societal elite following broader European trends than evidence of a mass movement or invasion.

So, the answer to your question is that "The Celts" never "arrived" in Ireland. Rather, during the middle Iron Age the people at the top of Irish society, being part of a broader western European cultural sphere, began to display their status through the use of objects that were made and decorated in the La Tène style, however, these objects were made locally, to suit local tastes and to work within already existing frameworks. For example, Irish La Tène horse gear has components which are entirely unique, and Irish La Tène swords are almost comically short compared to British or Continental examples.

Later, once Irish people start writing about themselves (really around 600 AD), we get these Irish writers self-identifying as Gaels, which seems to be used to identify people who speak the Irish language, the word itself deriving from the word used to describe them by people speaking Brythonic languages (Welsh).

This has been a really simple run down of a lot of things, any of which could have their own dedicated discussion, so if I haven't gone into enough depth at any stage feel free to ask about specific things.

Sources:

Collis. (2006) The Celts: Origins, Myths, & Inventions (2nd edition).

Raftery. (1994) Pagan Celtic Ireland

Raftery. (1983) La Tène in Ireland: Problems of origins and chronology

Edit: Found a citation that I couldn't remember off the top of my head

Koch (2003). Celts, Britons and Gaels: Names, peoples and identities. TRANSACTIONS-HONOURABLE SOCIETY OF CYMMRODORION, 41-56.

From the linguistic/historical side, you might find this previous answer of mine helpful as extra context to the post from u/Typologyguy about archaeology.