Learn Polish: Wczoraj był poza domem. - Yesterday he was outside the house.

The preposition poza requires the instrumental case when you are describing a static location or position (answering "where?"). Because the person simply "was" outside without any movement to a new place, we use the instrumental form domem.

Does poza domem mean physically outside the building, or just "away from home"?

It can mean both. While it can literally mean standing outside the house, poza domem is very often used as a set phrase meaning "away from home," "out of the house," or even "out of town."

Could I use na zewnątrz (outside) instead of poza in this sentence?

You could, but it changes the grammar and the exact focus. Na zewnątrz refers strictly to being physically outdoors and requires the genitive case (na zewnątrz domu). Poza is broader and simply means being "beyond" or "out of" the limits of the home.

Why is the word on (he) missing from the Polish sentence?

In Polish, the past tense verb był already tells us the subject is a single male (third-person singular masculine). Subject pronouns are usually dropped to sound more natural, unless you really need to emphasize exactly who was outside.