Learn Russian: Мне надо срочно сдать отчёт.

In Russian, expressing "I need to do something" uses an impersonal structure. Instead of a subject doing an action, you literally say "To me it is necessary." That is why we use the Dative case pronoun мне (to me) followed by надо (it is necessary) and the infinitive verb сдать (to hand in).

Could I use нужно instead of надо in this sentence?

Yes, absolutely! Мне нужно срочно сдать отчёт means exactly the same thing. Надо and нужно are completely interchangeable when followed by an infinitive verb, though надо can sometimes sound slightly more conversational.

I learned that сдать means "to pass an exam". Does it mean something else here?

Yes, сдать has a broader meaning of "to hand over" or "to submit". When you "pass" a written exam, you are essentially handing in your paper. Here, сдать отчёт simply means to hand in or submit the report. The perfective aspect is used because it refers to successfully completing this one specific action.

What case is отчёт in? It looks just like the dictionary form.

It is in the Accusative case because it is the direct object of the action (you are handing in the report). Because отчёт is a masculine inanimate noun (a thing, not a living person or animal), its Accusative form is identical to its Nominative (dictionary) form.