Learn German: Die Zustellung war binnen einer Woche erfolgreich. - The delivery was successful within a week.

It is actually ambiguous because the feminine forms for both dative and genitive are identical (einer Woche). The preposition binnen usually takes the dative case in standard German, but it is also widely used with the genitive in formal written German. Since both forms look the same for feminine nouns, either interpretation works perfectly here.

How would this sentence look if I used a masculine noun, like ein Monat (a month)?

Because masculine nouns have distinct dative and genitive endings, you would see the case difference clearly. You can say binnen einem Monat (dative, which is standard and very common) or binnen eines Monats (genitive, which is slightly more formal).

Could I use innerhalb instead of binnen to mean 'within a week'?

Yes, innerhalb einer Woche means the exact same thing. However, innerhalb is much more common in everyday spoken German. binnen sounds a bit more formal or bureaucratic, which perfectly matches a formal noun like Zustellung.

Why use Zustellung instead of Lieferung for 'delivery'?

Both translate to 'delivery', but Zustellung specifically refers to the formal act of handing over mail, parcels, or legal documents to a recipient (think of postal services). Lieferung is a broader term for the transport and supply of goods, like delivering bulk freight, a new sofa, or a pizza.