Two emotions are contrasted here, greann (humour) and fearg (anger)
The first phrase "an chiall atá ag Áine don ghreann" roughly translated means "the appreciation that Áine has for humour (fun?)" [= her sense of humour "She has a sense of humour" is usually "Tá féith an ghrinn inti."]
sin = é sin / sé sin = that
gheobhaidh sin an bua = that will overcome [that will get the victory]
ar a cuid feirge = over her anger
"cuid" is a word used to emphasise the possessive pronoun "a". Theoretically you could say "a fearg" but it wouldn't sound well. The use of "cuid" seems to be increasing. In my time it was only used of things held in common with others "mo chuid airgid" - "mo chuid gruaige" - and even "mo chuid" (my food)
There is a nice phrase associated with "cuid" -- "Tá mo chuid is mo bhuíochas agam anois." This is said when you offer something to someone and it is not required or the offer is not accepted. "I (still) have "mo chuid" (whatever I offered) and now I also have that person's gratitude "mo bhuíochas"."
I have received thanks for something I didn't even give! / Tá mo chuid is mo bhuíochas agam.
i gcónaí = always
a cuid feirge bréige = her false anger / her pretended anger
Translation: Her sense of humour will always overcome her pretended anger?
i.e. in a situation where she should be angry she will see the funny side
etc etc
May I ask where or in what book you found this sentence?