Learn Hardcore Finnish: Virkailija sanoo, että vuoronumero näkyy näytöllä pian. - The clerk says that the queue number will appear on the screen soon.

In Finnish, a subordinate clause introduced by että (that) is normally separated from the main clause with a comma.
So Virkailija sanoo, että … is the standard punctuation pattern: main clause + comma + että-clause.

What exactly does että do in this sentence?

Että introduces a content clause (a clause that contains what someone says/thinks/knows).
Here it marks the reported statement: the clerk says that the ticket number will be visible soon.

Why are both verbs in the present tense (sanoo, näkyy) if the meaning is about the future?

Finnish often uses the present tense for near-future situations when the timing is clear from context or an adverb like pian (soon).
So näkyy … pian literally looks like present tense, but it naturally means will be visible soon.

Is näkyy passive? It feels like English is shown.

Näkyy is not passive here. It is the 3rd person singular present of näkyä = to be visible / to show (in the sense of appear, be displayed).
Finnish expresses the idea without an agent:

vuoronumero näkyy näytöllä = the ticket number is visible on the screen / appears on the screen

If you specifically wanted a passive meaning like is being shown (by someone/system), Finnish might use:

vuoronumero näytetään näytöllä pian (passive of näyttää)

What case is näytöllä, and why is it used?

Näytöllä is the adessive case (-llä/-llä), often meaning on or at something.
So:

näyttö = a display/screen
näytöllä = on the screen

It answers the question where? (on what surface / at what location).

Why is it näyttö but näytöllä (only one t)?

That’s consonant gradation. The stem of näyttö changes in some forms:

strong grade: näyttö
weak grade used before many endings: näytö-

Then add -llä:

näytö + llä → näytöllä

So the tt → t change is normal and very common.

What is vuoronumero—is it one word or two?

It’s a compound noun written as one word (very common in Finnish):

vuoro = turn/shift
numero = number
vuoronumero = queue number / ticket number

Compounds in Finnish are usually written together rather than as separate words.

Why doesn’t Finnish use an article like the in vuoronumero?

Finnish has no articles (a/an/the). Definiteness is inferred from context.
Here it’s clear you mean the relevant queue number for the situation, so vuoronumero can correspond to the ticket number without any extra word.

Why is virkailija in the basic form (nominative)? Could it be something else?

Virkailija is the subject of sanoo, so it appears in the nominative (dictionary form).
If you changed the structure, the form could change, e.g. in some other constructions or if it were an object, but here it’s simply The clerk says…

Could the word order change, especially where pian goes?

Yes, Finnish word order is fairly flexible, but different placements can shift emphasis. Common options include:

… näkyy näytöllä pian. (neutral: soon)
… näkyy pian näytöllä. (slightly more focus on soon)
Pian vuoronumero näkyy näytöllä. (fronting pian for emphasis)

All are grammatical; the most “default” sounding one often keeps time adverbs like pian near the end.