In her pronunciation of phríomhchathair, ph sounds like an F, mh sounds like a V, and th sounds like a H, so “freev-kha-hair” would be an approximate English pronunciation. (The broad Irish ch is the same sound as at the end of German Bach — a bit throaty. There is some variation in the pronunciation of an Irish slender r ; sometimes it sounds something like a D, and sometimes it sounds something like a Z, but the sound isn’t found in English. One description of how to pronounce it can be found here.)
"Capitol" and "capital (city)" are two different things (and 'capital' seems to be accepted).
A capital is the official chief city of a country; a capitol is a building or complex of buildings within which governing functions are conducted. The former is just from the Latin for "head"; the latter is from the Latin name for the Capitoline Hill, where the Temple of Jupiter was built.
In the American context, the capital is Washington DC, but the Capitol is specifically the building in which Congress sits. The term is particularly used in the US because of its ideology as 'the New Rome'.
Most capitals contain a capitol, though they're not normally known by that name. There are exceptions, though: the capital of the Netherlands, for instance, is Amsterdam, but the capitol is in The Hague.