No, it is not technically a possessive pronoun. (Irish doesn't have possessive pronouns - did you mean a possessive adjective?).
The a linking a noun or pronoun to a verbal noun as in this exercise is a preposition. It only ever lenites, whether the noun is masculine or feminine, singular or plural, unlike the 3rd person possessive adjective.
The vocative particle a is also not a possessive adjective, and neither is the relative particle a, or the other different as that occur in Irish.