V té tiskárně už není papír, proto musím jít do obchodu.

Because the preposition v meaning in requires the locative case when it expresses location.

So:

• ta tiskárna = that printer (dictionary/basic form)
• v té tiskárně = in that printer

Both words change:

• ta → té
• tiskárna → tiskárně

This is very normal in Czech: prepositions often force a noun into a particular case, and adjectives/demonstratives must match it.

Tiskárně is locative singular.

The noun tiskárna is a feminine noun ending in -a, and after v for location, it goes into the locative:

• tiskárna → v tiskárně

The ending -ě is a common locative singular ending for many feminine nouns of this type.

So:

• tiskárna = printer
• v tiskárně = in the printer

And with the demonstrative:

• v té tiskárně = in that printer

Here už means something like any more / no longer / already, depending on how English phrases it.

In this sentence:

• už není papír = there is no paper any more / there is no longer any paper

Czech už often marks a change of situation:

• before, there was paper
• now, there isn’t

That is why English often translates it with any more or no longer, even though už by itself is also often taught as already.

Není papír is a natural way to say there is no paper.

A few useful points:

• papír here is treated as a mass noun, like paper in English.
• So Czech uses singular: papír
• není is singular too.

So:

• není papír = there is no paper
• nejsou papíry would mean something more like there are no papers/sheets/documents

You could also say není tam papír if you want to emphasize there, but in this sentence v té tiskárně already gives the location, so tam is unnecessary.

Czech has no articles like English a/an/the.

So papír can mean:

• paper
• the paper
• a paper

The exact meaning depends on context.

In this sentence, because we are talking about paper inside a printer, the meaning is naturally:

• there is no paper

not usually there is no piece of paper.

In this kind of existential sentence, the thing that exists or does not exist is often in the nominative:

• Je tam papír. = There is paper there.
• Není tam papír. = There is no paper there.

So in:

• V té tiskárně už není papír

the location is marked by v + locative (v té tiskárně), while the thing being talked about, papír, stays in the nominative.

Proto means therefore / so / for that reason.

In this sentence:

• V té tiskárně už není papír, proto musím jít do obchodu.
• There is no paper in that printer any more, so I have to go to the store.

Do not confuse it with protože, which means because.

Compare:

• Proto musím jít do obchodu. = Therefore / So I have to go to the store.
• Musím jít do obchodu, protože v tiskárně není papír. = I have to go to the store because there is no paper in the printer.

So:

• proto = result
• protože = cause

Yes. Musím jít means I must go or I have to go.

It uses:

• musím = I must / I have to
• jít = to go

Czech commonly uses a conjugated modal verb + infinitive:

• musím jít = I have to go
• můžu jít = I can go
• chci jít = I want to go

Unlike English, there is no word like to before the second verb:

• English: I have to go
• Czech: musím jít

Because do means movement to/into, while v means location in.

So:

• jít do obchodu = to go to the store
• být v obchodě = to be in the store

This is a very important Czech pattern:

• do + genitive = motion toward/into
• v + locative = location in

So the sentence says:

• musím jít do obchodu = I have to go to the store

not

• musím jít v obchodě, which would be incorrect.

Obchodu is the genitive singular of obchod.

That is because the preposition do requires the genitive:

• obchod = store/shop
• do obchodu = to the store / into the store

This pattern is extremely common:

• do domu = to the house
• do školy = to school
• do práce = to work

So here:

• jít do obchodu = go to the store

Yes, Czech word order is flexible, and other versions are possible.

The given sentence:

• V té tiskárně už není papír, proto musím jít do obchodu.

is natural and neutral.

But you could also hear:

• Papír už v té tiskárně není, proto musím jít do obchodu.
• Už není v té tiskárně papír, proto musím jít do obchodu.

These all mean roughly the same thing, but the emphasis changes slightly.

The original version starts with the location:

• V té tiskárně...
• This sets the scene: in that printer...

That is a very common Czech style.

Jít means to go on foot or more generally to go in many everyday contexts.

Jet usually means to go by vehicle.

So:

• musím jít do obchodu = I have to go to the store
• musím jet do obchodu = I have to go to the store by car/bus/etc.

In ordinary speech, jít do obchodu is very common unless the speaker wants to emphasize traveling by vehicle.

In modern everyday Czech, tiskárna commonly means printer.

Depending on context, it can also refer to a printing house / print shop, but in this sentence that would not fit well, because:

• V té tiskárně už není papír clearly suggests paper inside a machine

So here tiskárna definitely means printer.

Yes, absolutely.

• V té tiskárně už není papír.
• V té tiskárně už není žádný papír.

Both are correct.

The version with žádný is more explicit:

• žádný papír = no paper / not any paper

The shorter version without žádný is very natural and common in Czech, especially when the meaning is obvious from context.

So:

• není papír = natural, concise
• není žádný papír = more emphatic, explicitly no paper at all