כבר חודש אני לומדת עברית בספרייה אחרי העבודה.

Yes, כבר often means already, but in expressions like כבר חודש it means something like for a month now or already a month now.

So כבר חודש אני לומדת... gives the idea that the action started in the past and is still continuing now.

This is different from a more neutral duration phrase like במשך חודש, which simply means for a month.

• כבר חודש = ongoing up to now, with a sense of by now / already
• במשך חודש = duration, more neutral

Because the speaker is feminine singular.

In Hebrew present tense, verbs agree with the subject in gender and number:

• אני לומדת = I am studying / I study (female speaker)
• אני לומד = I am studying / I study (male speaker)

This is important because אני itself does not show gender. The verb does.

Hebrew normally does not use a separate present-tense verb meaning am / is / are in sentences like this.

So:

• אני לומדת can mean I study
• and also I am studying

Hebrew uses the same present-tense form for both ideas, and context tells you which is meant. In this sentence, the time phrase כבר חודש makes it clear that this is an ongoing situation.

Usually, in a normal standalone sentence, אני is kept.

That is because Hebrew present-tense verb forms show gender and number, but not person clearly enough on their own:

• לומדת means studying or [she / I feminine] study/am studying, depending on context

So אני לומדת is the normal, clear form for I am studying.

You can leave אני out in certain contexts, especially in casual speech, but then it usually depends on what was said before.

Because את is only used before a definite direct object.

Here, עברית means Hebrew in a general sense, not the Hebrew, so it is not definite.

Compare:

• אני לומדת עברית = I study Hebrew
• אני לומדת את העברית של התנ״ך = I study the Hebrew of the Bible

So no את is needed in your sentence.

בספרייה contains the preposition ב־ meaning in, attached directly to the noun.

Hebrew often attaches short prepositions directly to the following word:

• ב־ = in
• ל־ = to
• כ־ = like / as

When the noun is definite, the definite article ה־ is not written as a separate word after these prepositions the way English uses the. It gets absorbed into the form.

So בספרייה means in the library.

In everyday unpointed Hebrew spelling, the same written form can sometimes represent both meanings.

So in plain writing, בספרייה may be understood from context as:

• in the library
• or in a library

Usually context makes it obvious. If vowel marks were added, the distinction would be clearer in pronunciation:

• ba- often signals in the
• be- often signals in a

In your sentence, the intended meaning is clearly in the library.

Because Hebrew often uses the definite article in places where English does not.

אחרי העבודה literally means after the work, but in natural English we usually say after work. In Hebrew, the definite form is very normal here because it refers to the speaker’s workday or job in a familiar, specific sense.

So:

• אחרי העבודה = after work / after the workday
• אחרי עבודה sounds less natural for this everyday meaning and can feel more abstract or less idiomatic

Yes, it could come later. Hebrew word order is flexible.

These are all possible:

• כבר חודש אני לומדת עברית בספרייה אחרי העבודה
• אני לומדת עברית בספרייה אחרי העבודה כבר חודש
• אני לומדת עברית כבר חודש בספרייה אחרי העבודה
  (possible, though less neat)

Starting with כבר חודש gives extra emphasis to the time span. It is a very natural way to frame the sentence: for a month now...

It can mean either one.

Hebrew present tense usually does not distinguish between:

• simple present: I study
• present progressive: I am studying

So אני לומדת עברית can mean both, depending on context.

In this sentence, because of כבר חודש, English would often prefer something like:

• I’ve been studying Hebrew for a month now
• or For a month now, I’ve been studying Hebrew...

That ongoing meaning comes from the whole sentence, not from a special verb form.