The default rule is that, if you are resident or ordinarily resident in Ireland, you are liable to CAT on your worldwide gifts and inheritances.
That rule can be modified in relation to a particular country by the terms of a Double Taxation Agreement between Ireland and that country. However most of Ireland’s DTAs say nothing about inheritance tax/estate tax/CAT. (This is because they are based on an OECD model tax treaty that does not cover inheritance tax/estate tax/CAT.)
But Ireland’s DTAs with the US and the UK do address inheritances. Under the IRL/US DTA federal estate tax paid in the US is allowable as a credit against CAT liability in Ireland. That wouldn’t help here because, apparently, no US estate tax is payable, so no credit.
But, In addition, the DTA provides that, if the deceased was domiciled in the US and not in Ireland (which, it seems, she was) then the inheritance of property located in the US does not attract CAT in Ireland. There are complex rules in the DTA for determining where property is located.
In this case we’re told that the Irish resident is inheriting a “sum of money”. If that’s sitting in a bank account belonging to the deceased, then I think there’s no problem. But if it represents, say, the value of the deceased’s shares in an Irish company (or, indeed, a company incorporated anywhere outside the US), then that’s not property situated in the US, and so CAT is payable.
So it crucially matter exactly what property was left under the terms of the will. There’s an excellent chance that it doesn’t attract CAT but, if it turns out to be the wrong kind of property, it may.
SFAIK even if there is no liability to CAT you still have to file a CAT return disclosing the inheritance and claiming the relief.