Translation:I broke up with my boyfriend last month.
The grammar's a bit wonky in your sentence.
Your friend in Japan is apparently female (you say "she told me"). When you say "him", you must be referring to your classmate (who must then be male). Your classmate and your classmate's girlfriend visited your Japanese friend. But you say "my classmate's gf [...] broke up with her". I assume what you meant was that your classmate broke up with his girlfriend, not that the girlfriend broke up with herself.
It depends on the context, from my understanding. The gatsu is used mostly when referring to months by their names. Nigatsu - Feb, sangatsu - march. And then getsu is used for all other occasions, like the one used here - sengetsu. This explanation is a generalization though - always exceptions.