yes, it is an aspiration, an aspiration is when there is a h after the second or third letter in certain words that makes it have a softer sound.
"Good" here is used as a qualitative adjective specifying which woman (the good one). "The woman is good/well," on the other hand, is specifying the condition of the woman (good-quality/well-condition) in predicative adjective/copula. I'd want to say (in probably awful beginner's grammar) "Tá an bean go maith" ("the woman is good") or something along those lines would convey that the woman is doing well. (Note though that I'm still new at Irish and could be totally wrong; Irish is fairly different from Latin and Japanese grammar...)