Learn Russian: Я передаю ему письмо.

The verb передавать (to pass / to hand over) ends in -авать. In the present tense, these verbs drop the -ва-. So, instead of передаваю, it becomes я передаю.

Why is it ему instead of его?

Ему is the Dative case of он (he). We use the Dative case for the indirect object, which is the person receiving something. Here, you are passing the letter to him.

Is письмо in the Nominative or Accusative case here?

It is in the Accusative case because it is the direct object (the thing being passed). However, for neuter nouns like письмо, the Accusative form looks exactly the same as the Nominative (dictionary) form.

How would I change this sentence to politely say 'I would like to pass him the letter'?

You would use the phrase from this lesson: Я хотел бы (if you are male) or Я хотела бы (if female), followed by the infinitive. So it becomes: Я хотел бы передать ему письмо. Notice we use the perfective infinitive передать here because it is a single, completed action.