Learn Italian: Il litigio era sul punto di finire.

The phrase essere sul punto di describes a state of being 'about to' do something. In the past, this is almost always a background condition or an ongoing state, which requires the imperfect tense (era). Using the passato prossimo would sound unnatural because 'being about to' isn't a completed, contained action.

Does essere sul punto di finire mean the exact same thing as stare per finire?

They are very close synonyms and often interchangeable. However, essere sul punto di (literally 'to be on the point of') feels slightly more dramatic or imminent than stare per. It emphasizes the very last second before an action happens.

I always want to translate 'argument' as argomento. Why is it litigio here?

Argomento is a very common false friend that actually means 'topic', 'subject', or 'theme'. When you mean a verbal fight or quarrel, you need to use words like litigio, lite, or discussione.

Can I change the preposition and say sul punto a finire since it translates to 'about TO end'?

No, the phrase is strictly fixed as essere sul punto di + infinitive. Even though it translates to 'about to' in English, you must always use the preposition di in Italian for this specific construction.