Геодезический прибор уже работает.

In Russian, the present-tense form of to be is usually omitted.

So:

• Геодезический прибор уже работает
  literally looks like
• Geodetic instrument already works / is working

Russian does not normally say a separate word for is here.

It is in the nominative singular.

Why?

• прибор is the subject of the sentence.
• Subjects are normally in the nominative case.
• геодезический matches прибор in case, number, and gender.

So:

• геодезический = nominative singular masculine
• прибор = nominative singular masculine

The word уже is an adverb, so it does not change for case.

Because it is a masculine singular nominative adjective, agreeing with прибор.

Russian adjectives change their endings to match the noun they describe.

Here:

• прибор is masculine
• singular
• nominative

So the adjective also takes the masculine singular nominative form:

• геодезический прибор

If the noun changed, the adjective would change too. For example:

• геодезическая система = a geodetic system
• геодезическое оборудование = geodetic equipment

Yes, прибор is masculine.

A good first clue is that many Russian nouns ending in a consonant are masculine.
Since прибор ends in -р, it is usually treated as masculine.

You can also see this from the agreement:

• геодезический uses a masculine form
• работает is the same for all genders in the present tense, so the adjective is the clearer clue here

Уже usually means already.

In this sentence, it shows that the instrument has reached the state of working/operating by now.

So the sentence suggests something like:

• it was not working before, but now it is
• or the speaker expected this change and is noting that it has happened

It is a very common adverb in Russian.

It can mean both, depending on context.

Russian present tense often covers both:

• simple present: works
• present progressive: is working

So прибор работает can mean:

• the instrument works
  or
• the instrument is working

In this sentence, because of уже, English will often use is already working or already works, depending on the situation.

Yes, absolutely. Russian very commonly uses работать for machines, devices, systems, engines, programs, and so on.

So прибор работает means:

• the instrument is working
• the instrument is operating
• the device is functioning

This is completely natural Russian.

Работает is:

• 3rd person singular
• present tense
• from the verb работать = to work

It agrees with a singular subject:

• прибор работает = the instrument works / is working

A quick comparison:

• я работаю = I work / am working
• ты работаешь = you work / are working
• он / она / оно работает = he / she / it works / is working

Russian uses the imperfective verb работать in the present tense because it describes an ongoing state or process: the device is functioning.

Perfective verbs normally do not have a true present tense. Their “present” forms usually refer to the future.

So here the imperfective is the natural choice:

• работает = is working / works

That is exactly what you want for a current situation.

Yes, Russian word order is flexible, but the most neutral order here is:

• Геодезический прибор уже работает.

This sounds like a normal statement: subject + adverb + verb.

Other orders are possible, but they change emphasis:

• Уже работает геодезический прибор.
  More marked; it emphasizes already working first.

• Геодезический прибор работает уже.
  Possible in some contexts, but less neutral here and often sounds incomplete unless something follows.

So for a learner, the original order is the safest and most natural.

Russian has no articles.

So прибор can mean:

• a device / an instrument
• the device / the instrument

The exact meaning depends on context.

That means Геодезический прибор уже работает could be understood as either:

• The geodetic instrument is already working
• A geodetic instrument is already working

Usually the broader context tells you which one is meant.

A helpful stressed version is:

Геодези́ческий прибо́р уже́ рабо́тает.

Approximate pronunciation:

• геодези́ческий = gyee-uh-dye-ZEE-chi-skiy
• прибо́р = pree-BOR
• уже́ = oo-ZHE
• рабо́тает = ra-BO-ta-yet

A few useful notes:

• Russian stress is important and can move from word to word.
• Unstressed vowels are often reduced, so the actual spoken form may sound less like the spelling than in English.

It can often be translated as geodetic, and in many practical contexts also as surveying.

So геодезический прибор may be translated as:

• geodetic instrument
• surveying instrument

The best English choice depends on the context and on how technical the text is. In everyday translation, surveying instrument may sound more natural; in technical or scientific writing, geodetic instrument may fit better.