Манометр уже работает.

It is in the nominative case because it is the subject of the sentence — the thing that is doing the action of работает.

So:

• манометр = nominative singular
• it is a masculine noun
• the sentence structure is basically [subject] + [adverb] + [verb]

Работает is the 3rd person singular present-tense form of работать.

It matches манометр, which is:

• singular
• he/it in grammatical agreement

So:

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

Also, работаеть is not a correct form. The infinitive is работать.

Yes. In Russian, the present tense often covers both:

• works
• is working

So Манометр уже работает can mean:

• the pressure gauge already works
• the pressure gauge is already working
• the pressure gauge is already operating

Russian does not usually need a separate form like English is working.

Уже usually means already.

Here it shows that the situation has changed or happened sooner than expected:

• before, it was not working
• now it is working

Depending on context, уже can also add a sense like:

• by now
• at this point
• now already

So the word is small, but it often carries the idea of a change of state or earlier-than-expected completion.

Yes, Russian word order is fairly flexible.

The neutral order here is:

• Манометр уже работает.

This is natural because:

• манометр is the topic/subject
• уже modifies the verb
• работает comes at the end

But other orders are possible for emphasis, for example:

• Уже манометр работает.
• Манометр работает уже.

These are possible in the right context, but they sound less neutral and more marked. For a learner, Манометр уже работает is the safest normal version.

Yes, манометр is masculine.

A common clue is that many Russian masculine nouns in the dictionary form end in a consonant, and манометр ends in -р.

That is why you would say:

• новый манометр = a new pressure gauge
• этот манометр = this pressure gauge

If it were feminine or neuter, the adjective and pronoun forms would be different.

The stress is:

• мано́метр
• уже́
• рабо́тает

Stress matters in Russian, because it is not always predictable. A learner should try to remember the stressed syllable together with each new word.

Yes, absolutely. Работать is very commonly used for:

• people
• machines
• devices
• systems
• programs
• methods

So манометр работает is completely natural and means that the device is functioning.

Examples:

• Телефон работает. = The phone works / is working.
• Система уже работает. = The system is already working.
• Я работаю. = I work / am working.

Russian does not have articles like English a or the.

So манометр can mean:

• a pressure gauge
• the pressure gauge

The exact meaning depends on context.

In many real situations, Манометр уже работает would naturally be understood as The pressure gauge is already working, because the speaker and listener already know which one they mean.

A very common way is:

• Манометр ещё не работает.

This means The pressure gauge is not working yet.

Notice the difference:

• уже = already
• ещё не = not yet

Compare:

• Манометр уже работает. = The pressure gauge is already working.
• Манометр ещё не работает. = The pressure gauge is not working yet.

It is a perfectly complete sentence on its own, but in real life it often feels like part of a situation where someone was waiting for the device to start functioning.

Because of уже, it often implies a background such as:

• someone repaired it
• someone switched it on
• someone was checking whether it had started working

So grammatically it is complete, but pragmatically it often suggests some earlier expectation or delay.