Learn Hardcore Norwegian: Vaktmesteren sier at hun kan komme innom i morgen og se på døren. - The caretaker/janitor says that she can stop by tomorrow and look at the door.

Vaktmesteren is the definite form: vaktmester + -en = the caretaker/janitor/superintendent (a specific, known one).
If you said en vaktmester, that would mean a caretaker (not previously identified).

What does vaktmester actually mean in Norwegian housing contexts?

Vaktmester is the person responsible for maintenance in a building or housing complex—often translated as caretaker, superintendent, building maintenance person, sometimes janitor depending on context. It’s a broad role: fixing things, checking doors/locks, coordinating repairs, etc.

Why does Norwegian use sier at (that) here—can I omit at?

After verbs like å si (to say), Norwegian commonly uses at to introduce the clause, like English that.
You can sometimes omit at in informal speech, but:

Vaktmesteren sier at hun kan komme … is very standard and clear.
Omitting at is more colloquial and can make longer sentences harder to read.