I believe that the breviary approved for use in Ireland is the three-volume Divine Office published by HarperCollins, Volume 1 of which is visible in the link provided by Dcn. Patrick. UK price: £50 per volume (it’s leather bound).
Safest bet is to get whatever your priest is using.
Deacon Patrick is correct – the version you are looking at, Alexiana, is the version of the Americans and so it would have their calendar for feasts and memorials, which would put you at a disadvantage. Also quite a number of their texts are rendered into their translation of English, most notably the Gospel canticles that you would be reciting daily at morning prayer, evening prayer, and night prayer. So you would be using texts not normative to the liturgy where you are.
UpUpAndAway is right that the link that the Deacon gives has the version used in Ireland. For one who is not obliged to the office, you could of course get the American version but it is awkward when you say it with anyone else who will have the other version – and you would miss celebrations because of having the wrong national calendar.
To give you a sense of how the texts vary:
The American version of the Canticle of Mary:
My + soul proclaims the greatness of the Lord, †
my spirit rejoices in God my Savior *
for he has looked with favor on his lowly servant.
From this day all generations will call me blessed: †
the Almighty has done great things for me, *
and holy is his Name.
He has mercy on those who fear him *
in every generation.
He has shown the strength of his arm, *
he has scattered the proud in their conceit.
He has cast down the mighty from their thrones, *
and has lifted up the lowly.
He has filled the hungry with good things, *
and the rich he has sent away empty.
He has come to the help of his servant Israel *
for he has remembered his promise of mercy,
the promise he made to our fathers, *
to Abraham and his children for ever.
Glory to the Father, and to the Son, *
and to the Holy Spirit:
as it was in the beginning, is now, *
and will be for ever. Amen.
Which is different from:
The Magnificat
My soul glorifies the Lord,
My spirit rejoices in God my Saviour.
He looks on his servant in her lowliness;
Henceforth all generations will call me blessed.
The Almighty works marvels for me.
Holy his name!
His mercy is from age to age,
on those who fear him.
He puts forth his arm in strength
And scatters the proud hearted.
He casts the mighty from their thrones
And raises the lowly.
He fills the starving with good things,
Sends the rich away empty.
He protects Israel, his servant,
remembering his mercy,
the mercy promised to our fathers,
to Abraham and his sons for ever.
Glory be to the Father and to the Son
and to the Holy Spirit,
as it was in the beginning, is now,
and ever shall be, world without end.
Amen.
