Locuiesc în acest apartament de mult timp.

Locuiesc is the 1st person singular present tense form of the verb a locui, which means to live / to reside.

So:

• a locui = to live, to reside
• locuiesc = I live

In this sentence, locuiesc refers to where the speaker lives.

Romanian often leaves out subject pronouns when they are already clear from the verb ending.

Here, locuiesc already means I live, so eu is not necessary.

• Locuiesc în acest apartament... = I live in this apartment...
• Eu locuiesc în acest apartament... = also correct, but more emphatic, like I live in this apartment...

You usually add eu only if you want contrast, emphasis, or clarity.

Locuiesc is in the present tense.

Romanian commonly uses the present tense for something that started in the past and is still true now, especially with expressions like de mult timp.

So this Romanian present tense can correspond in English to:

• I live in this apartment
• I have lived in this apartment for a long time
• I have been living in this apartment for a long time

English often prefers the present perfect here, but Romanian normally uses the simple present.

În means in.

It is the normal preposition used for being inside a place or location:

• în casă = in the house
• în oraș = in the city
• în acest apartament = in this apartment

So Locuiesc în acest apartament literally means I live in this apartment.

Acest means this, and it agrees with apartament.

Apartament is a neuter noun in Romanian. In the singular, neuter nouns behave like masculine nouns, so you use acest:

• acest apartament = this apartment

Some related forms:

• acest = this (masculine/neuter singular)
• această = this (feminine singular)
• acești = these (masculine plural)
• aceste = these (feminine/neuter plural)

Because apartament is singular neuter, acest is the correct form.

Yes. Romanian neuter nouns are a special category:

• in the singular, they behave like masculine
• in the plural, they behave like feminine

So:

• acest apartament = this apartment
• două apartamente = two apartments
• aceste apartamente = these apartments

That is typical neuter behavior in Romanian.

De mult timp means for a long time or for a long time now.

Breakdown:

• de = for / since, in time expressions
• mult = much, a lot
• timp = time

Together, de mult timp is an idiomatic expression meaning that something has been going on for a long period.

In this sentence, it tells you how long the speaker has lived there.

In Romanian, de is commonly used to express duration with ongoing actions or states.

So:

• Locuiesc aici de doi ani. = I have lived here for two years.
• Te aștept de o oră. = I have been waiting for you for an hour.
• Locuiesc în acest apartament de mult timp. = I have lived in this apartment for a long time.

This is a very common Romanian pattern:

present tense + de + time expression

Yes. Romanian word order is fairly flexible.

You can also say:

• Locuiesc de mult timp în acest apartament.

This means the same thing. The difference is mainly one of focus or rhythm:

• Locuiesc în acest apartament de mult timp.
  Focuses first on where you live.
• Locuiesc de mult timp în acest apartament.
  Brings the idea of duration a bit earlier.

Both are natural.

Yes. Both mean this apartment:

• acest apartament
• apartamentul acesta

The first puts the demonstrative before the noun; the second puts it after the noun, which is also very common in Romanian.

Very roughly:

• acest apartament can sound a bit more neutral or formal
• apartamentul acesta can sound a bit more conversational or pointed

In your sentence, either is possible:

• Locuiesc în acest apartament de mult timp.
• Locuiesc în apartamentul acesta de mult timp.

Both are correct.

You could say:

• Eu locuiesc în acest apartament de mult timp.

This is still correct, but now eu adds emphasis. It may suggest contrast, such as:

• Eu locuiesc în acest apartament, dar fratele meu locuiește în altul.
  I live in this apartment, but my brother lives in another one.

Without emphasis, Romanian usually prefers simply:

• Locuiesc în acest apartament de mult timp.