So I've noticed that in the (Munster) Irish that I'm learning, we the number and person of a verb is indicated by the suffix, rather than a separate pronoun after the verb. This, I believe, is an example of a synthetic feature of the Irish language. In the Rosetta stone I got from my local library, the number and person are indicated by a pronoun after the verb rather than a suffix being added to the verb.

I read online that Celtic languages are "synthetic" languages, which makes me think that 400 years ago it was probably rare in Ireland for the verb and pronoun to be spoken separate (except in certain cases, like 2nd person plural for example).

Sorry if dumb question

Without being a pro on this, I'd assume that synthetic forms were more common throughout the country the further in time you go back - as, from what I know, Munster Irish is held as grammatically the most conservative of the dialects.
As an aside - this is funny in a way as Munster Irish is phonetically the least conservative - with Donegal Irish keeping the most phonemes.

Outside of Munster, there are still plenty of synthetic forms heard in the tenses of Galway Irish - I've heard it in the past habitual and conditional at least.
Equally I've heard Galway speakers use bhíodar and sometime bhíos in the past tense.

Even in Munster there's some variation on synthetic forms between the future and present tense - see this thread for some good detail from An Lon Dubh.

I dare say younger Munster native speakers will use less and less of these forms, with the future then past tense starting to go first.
From what I've heard, you have a situation where you have teachers with poor Irish teaching government "standard" forms, overwriting the traditional (and correct) local forms - Rosetta Stone probably does the same, teaching "standard" Irish.

Yes, older forms of Irish had significantly more verbal inflection for subject. In fact, most Indo-European languages did, if you go back far enough.

The Munster dialect has preserved more verbal morphology than other dialects, though, subject inflection still occurs in all dialects in certain persons, tenses and moods.