Learn Arabic (Modern Standard): أخبرتني صديقتي أن الخبر صدمة. - My friend f told me that the news was a shock.

This is exactly how sequence of tenses works in Arabic reported speech! In English, we backshift the tense to the past because the telling happened in the past. Arabic keeps the tense of the original statement. The friend originally thought or said 'The news is a shock' (a present-tense nominal sentence with no verb). So, Arabic just uses أن (that) followed by the original verbless sentence: الخبر صدمة.

Why do we use أن here instead of إنّ for 'that'?

We use إنّ almost exclusively after the verb قال (to say) and its variations. For other verbs of communication, feeling, or thought, such as أخبر (to tell) in this sentence, we must use أن.

Why does the object pronoun for 'me' come before the subject 'my friend' in أخبرتني صديقتي?

In Arabic, when the direct object is a pronoun attached to the verb (like ني for 'me' in أخبرتني), it physically attaches to the verb and therefore must come before the standalone subject noun صديقتي (my friend).

If I were pronouncing the case endings, what vowel would go on the end of الخبر?

It would take a fatha (an 'a' sound), making it الخبرَ. This is because أن acts just like إنّ (they are 'sisters'): it forces the subject of the following nominal sentence into the accusative case.