Translation:Slàinte a Ruairidh.
The single letter "a" in Gaelic can mean several things depending on the context. I don't think "this" is one of them though. In this case it is simply part of the way the vocative case is formed in Gaelic, and it is known in this context as the vocative particle (participle?). There's quite a detailed explanation in the tips for the "Names" skill.
It cannot mean 'this' but it can mean 'that' (relative), which might be causing confusion:
An duine a chunnaic mi 'the person that saw me' or 'the person that I saw' (ambiguous)
No it's not a participle. That is a word like briste, 'broken'.
The link for this section of notes is https://duome.eu/tips/en/gd#Phrases
I understand your frustration. I am American. I do not use vocative embellishments when talking to Ruaraidh. Nouns do not have gender. Both of these things, that I do not understand, come into play here. The only answer that I have worked out to why these thing are is this:
They just are.
I hope this helps.
There are two different things here:
This is caused by the vocative particle a. In Gaelic, there is no lenition shown in writing on l, n or r - i.e. they do not add an h. There may be a very slight lenition heard but nothing to worry about. You also get no lenition at all when s is followed by a consonant other than these three, such as sg or st.
This is marked in masculine singular words only with an i before the final consonant. But Ruairidh already has one, so nothing doing there.