Learn Finnish: He seurustelivat kauan, mutta ero tapahtui kuukausi sitten. - They dated for a long time, but the break-up happened a month ago.

When expressing that something happened exactly one unit of time ago, the time word stays in the nominative (basic) form. If the number was more than one, you would use the partitive case, for example kaksi kuukautta sitten (two months ago).

Since the lesson also covers "kuluttua", could I use it in this sentence instead?

No, because sitten specifically means "ago" for events in the past. The postpositions kuluttua and päästä are used for the future, meaning "in" or "after" a certain period of time.

Why do we use "kauan" here instead of a word like "pitkä"?

Pitkä is an adjective describing the physical length of a noun or a long period as a concept (like pitkä aika). Kauan is an adverb specifically meaning "for a long time", and it is used here because it describes how long the verb action (seurustelivat) took place.

Does the word "ero" only mean a break-up in Finnish?

While it means "break-up" or "divorce" in this relationship context, ero is a versatile word that also commonly means "difference" (between two things) or "resignation" from a job.