Learn Hardcore Icelandic: Hún athugar póstnúmerið vandlega áður en hún sendir pakkann. - She checks the postal code carefully before she sends the package.

Yes. Icelandic usually attaches the definite article to the end of the noun as a suffix.
póstnúmer (postal code, neuter) → póstnúmerið (the postal code).

A quick snapshot (singular):

Indefinite: póstnúmer = a postal code
Definite: póstnúmerið = the postal code

Because it’s neuter, the definite ending in the singular is commonly -ið (or -ið/-ð depending on the word).

Why is póstnúmerið the same form in the sentence—shouldn’t it change case?

It is in a case (it’s the direct object of athugar), but for many neuter nouns the nominative and accusative singular look identical.
So póstnúmerið can be either nominative (subject) or accusative (object) by form alone—you tell from the sentence structure.

Here it’s accusative because:

Hún athugar [póstnúmerið] = She checks [the postal code]

Why does pakkann end in -ann?