Learn Hardcore German: Der Notarzt bringt den Patienten in die Notaufnahme, damit er nicht lange warten muss. - The emergency doctor brings the patient to the emergency room so that he doesn’t have to wait long.

Because der Notarzt is the subject of the main clause—the person doing the action (bringt = brings). Subjects are in the nominative case, and der is the nominative masculine definite article.

Why is it den Patienten and not der Patient?

Den Patienten is the direct object—the person being brought. Direct objects take the accusative case.

nominative: der Patient
accusative: den Patienten
Also, Patient is a masculine noun, and many masculine nouns referring to people belong to the n‑declension, so they often add -en in cases other than nominative: den Patienten, dem Patienten, etc.

How do I know that in die Notaufnahme uses die (accusative) instead of der (dative)?

With in, German chooses case based on meaning:

accusative = movement/direction (into somewhere)
dative = location (in/at a place)

Here, bringt ... in die Notaufnahme implies movement into the ER, so it’s accusative: in die Notaufnahme.
If it were location, you’d get dative: in der Notaufnahme (= in the ER).

What’s the grammar behind damit er nicht lange warten muss?

Damit introduces a purpose clause (roughly so that / in order that). In a subordinate clause introduced by damit, the conjugated verb goes to the end.
That’s why you see: ... warten muss (with muss at the end).

Why does the sentence have two verbs at the end: warten muss?

Because müssen is a modal verb. With modal verbs, German typically uses:

the main verb in the infinitive (warten)
the modal as the conjugated verb (muss)

And in subordinate clauses, the conjugated verb (muss) goes at the very end: ... warten muss.

Who does er refer to here—the emergency doctor or the patient?

In context, er almost certainly refers to den Patienten (the patient), because the idea is: the doctor brings the patient to the ER so that the patient doesn’t have to wait long.
German pronouns usually refer to the most relevant/likely antecedent, even if it’s in a different case (accusative here). If there were any ambiguity, German might repeat the noun (damit der Patient ...) to be extra clear.

Is nicht lange the only way to say not for long / not long here?

It’s a very common and natural way. Alternatives exist, but they shift emphasis:

damit er nicht lange warten muss = so he doesn’t have to wait long
damit er nicht so lange warten muss = so he doesn’t have to wait that long (stronger contrast)
damit er nicht lange warten braucht is possible, but muss is more straightforward and common in this type of statement.

Why is the word order Der Notarzt bringt den Patienten ... and not something else?

German main clauses follow the verb-second (V2) rule: the conjugated verb (bringt) must be the second element.
Here: 1) Der Notarzt (first element)
2) bringt (second element)
Then come the other parts: den Patienten, in die Notaufnahme, and finally the purpose clause.

Could I also say Der Notarzt bringt den Patienten zur Notaufnahme? What’s the difference?

Yes, and it’s common. The difference is mainly nuance:

in die Notaufnahme = into the ER (focus on entering the department)
zur Notaufnahme (= zu der) = to the ER (focus on going there; arrival implied but “into” is less explicit)

With bringen, both can work depending on what you want to emphasize.

What exactly is Notarzt vs Arzt, and what is Notaufnahme?

Arzt = doctor (general)
Notarzt = emergency physician / emergency doctor (often specifically dispatched for emergencies)
Notaufnahme = emergency department / ER (literally emergency intake/admissions)

So the sentence describes emergency services bringing a patient directly to the ER to reduce waiting time.