"Tea! Thank you, sister!"
Translation:Tì! Tapadh leat a phiuthar!
A is the vocative particle. Whenever you are talking to someone starting with a consonant, you add a in front. It's a bit like saying oh friend in English. This a causes lenition (i.e. adding an h after the majority of consonants). It is not used this side of the Irish Sea in front of a vowel (or fh + vowel).
The 'a' here is the vocative particle. it tells us that the speaker is addressing their sister. The vocative case causes lenition, hence "piuthar" > "phiuthar".
This is one of the tricky ones we just have to get used to reading the context of. On it's own "a phiuthar" could be someone addressing their own sister, or it could be someone talking about their (male) friend's sister. "Taing, a phiuthar" is me thanking my sister, "thuirt mi sin ri a phiuthar" is me telling someone that "i told his sister that".
I am afraid I do not quite agree with you, either pedagogically or grammatically. Firstly, people do not usually get confused whether a vocative is being used, as it is usually obvious from the context, so I think mentioning the word for 'his' just introduces a complication where there is none.
But grammatically, this words is pretty tricky, with an irregular genitive, which may confuse about the dative - but in fact it is ri a phiuthair - i.e. the dative is formed regularly from the nominative, notwithstanding the genitive. This word is also pretty counter-intuitive, as it will probably be the only word you ever meet that is feminine and ends in -ar. Of course some - or all - of this may be something you haven't covered yet, as they have a deliberate policy of introducing things, such as the preposition here, with simple examples that happen not to require anything odd, before going onto more tricky examples later.