لا أعرف اسم هذه الجارة، لكنها تسكن فوق شقتي منذ شهر.

Because الجارة means the female neighbor. The masculine form is الجار. Since the sentence is talking about a woman, Arabic uses the feminine noun.

هذه is the demonstrative for a feminine singular noun: this.
هذا is used with masculine singular nouns.

So:

• هذه الجارة = this female neighbor
• هذا الجار = this male neighbor

The demonstrative has to match the gender of the noun.

In standard Arabic, when a demonstrative like هذا / هذه is followed by a noun, that noun is usually definite, so it normally takes الـ.

So Arabic prefers:

• هذه الجارة = this neighbor

rather than:

• هذه جارة

That second version is not the normal MSA pattern for this neighbor.

Because Arabic often expresses of by putting two nouns together in an iḍāfa structure (construct phrase).

So:

• اسم هذه الجارة literally = name this neighbor
• natural English = the name of this neighbor

In this structure:

• اسم = the first noun
• هذه الجارة = the thing it belongs to / is connected to

Arabic does not need a separate word like of here.

Because اسم is the first part of an iḍāfa. In an iḍāfa, the first noun normally does not take الـ and does not take tanwīn.

So Arabic says:

• اسم هذه الجارة

not:

• الاسم هذه الجارة

Even though the whole phrase is definite in meaning, the first noun stays without الـ because of the grammar of iḍāfa.

In Arabic, the verb itself often already includes the subject.

• أعرف = I know
• لا أعرف = I do not know

So there is no need to add a separate word for I unless you want emphasis. The subject I is understood from the verb form أعرف.

لكنها means but she or however, she.

It is made of:

• لكن = but / however
• ها = she / her (attached pronoun here meaning she as the subject of the next idea)

So:

• لكنها تسكن... = but she lives...

In fully vowelled writing, you may see it as لكنّها.

Because in the present tense, the 3rd person feminine singular usually takes تـ at the beginning.

So:

• هي تسكن = she lives
• هو يسكن = he lives

That is why the sentence uses تسكن for the female neighbor.

Because when a noun ending in ة takes a suffix like ـي (my), the ة changes to ت in spelling and pronunciation.

So:

• شقة = apartment
• شقتي = my apartment

This is a very common pattern:

• سيارة → سيارتي
• غرفة → غرفتي
• شقة → شقتي

Literally, it means above my apartment.

• فوق = above / over / upstairs from
• شقتي = my apartment

In this context, فوق شقتي means she lives in the apartment above mine.

With a present-tense verb, منذ شهر often means something like for a month now or since a month ago.

So in natural English, this sentence is best understood as:

• She has been living above my apartment for a month
• or She’s lived above my apartment for a month

Arabic uses منذ with time expressions to refer to a period starting in the past and continuing up to now.

Here شهر means a month as a time expression after منذ.

A few useful points:

• Arabic often uses the singular noun after منذ in expressions like منذ شهر = for a month / since a month ago
• In fully vowelled text, you might see a case ending: منذ شهرٍ
• شهور would mean months, which would change the meaning

So منذ شهر is the normal way to say for a month in this sentence.