Learn Russian: Я посмотрю фильм с ней.

The standard instrumental case form for "her" is ей. However, when third-person pronouns come directly after a preposition like с (with), Russian adds an н to the beginning to make it easier to pronounce. That is why we say с ней instead of just с ей.

What case is ней and why is it used here?

It is in the instrumental case. This lesson focuses on doing things "with" someone, which requires the preposition с followed by the instrumental case. This shows accompaniment, meaning you are watching the film together with her.

Since фильм is the thing being watched, why didn't its ending change?

фильм is indeed the direct object, so it is in the accusative case. But because it is a masculine inanimate noun (a non-living thing), its accusative form looks exactly the same as its basic dictionary form.

What is the difference between смотрю and посмотрю in this sentence?

смотрю is the present tense form, meaning "I watch" or "I am watching". By adding the prefix по-, the verb focuses on a completed action in the future. That is why я посмотрю translates to "I will watch" rather than "I am watching".