Learn Russian: Она чуть не пролила горячий кофе на мой новый проездной! - She almost spilled hot coffee on my new transit pass!

The phrase чуть не is a fixed construction used for imminent averted actions. Even though it contains the negative particle не, it means 'almost did something.' It emphasizes that a disastrous or sudden event was right on the brink of happening, but was narrowly avoided.

Does чуть не always have to be followed by a perfective verb like пролила?

Yes, it requires a perfective verb, nearly always in the past tense. Because you are talking about a specific, one-time result that almost crossed the finish line into reality but was ultimately averted, the perfective aspect is mandatory.

We are talking about spilling coffee on something. Why do we use the Accusative case (на мой новый проездной) instead of the Prepositional?

The verb пролить (to spill) implies directional motion of a liquid onto a surface. When на shows the destination or direction of an action rather than a static location, it requires the Accusative case.

The word проездной (transit pass) has an adjective ending. Is it a noun or an adjective?

It is a substantivated adjective, meaning it is an adjective functioning as a noun. It originally comes from the phrase проездной билет (transit ticket). Because of its form, you must decline it exactly like a standard Russian adjective.