Nejdřív musím napsat e-mail sekretářce.

Because napsat is the perfective verb: it means to write something and complete it.

• psát = to write, to be writing, writing as a process
• napsat = to write/finish writing

So Musím napsat e-mail means I have to write the email (and get it done).

If you said musím psát e-mail, it would sound more like I have to be writing an email or I need to spend time writing an email, with more focus on the activity than the result.

Musím is the 1st person singular form of muset, meaning must / have to.

So:

• musím = I must / I have to

A few useful forms of muset:

• musím = I must
• musíš = you must
• musí = he/she/it must
• musíme = we must
• musíte = you all / formal you must
• musí = they must

In this sentence, musím napsat = I have to write.

Because sekretářce is in the dative case.

The base word is sekretářka = secretary.
Here it means to the secretary, and Czech usually expresses that idea with the dative, not with a separate word like English to.

So:

• sekretářka = secretary
• sekretářce = to the secretary

This is a very common pattern in Czech: the person receiving something often goes in the dative.

Because Czech often uses case endings instead of separate prepositions for this meaning.

In English, you say:

• write an email to the secretary

In Czech, the recipient is shown by the dative case:

• napsat e-mail sekretářce

So sekretářce already contains the idea of to the secretary.

Here e-mail is the direct object, so it is in the accusative case.

But e-mail is a masculine inanimate noun, and for many masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: e-mail
• accusative: e-mail

That is why you do not see a visible change here.

No. In modern Czech, you may also see:

• e-mail
• email

Both are used. E-mail with the hyphen is very common and fully natural, but email is also widespread.

In a learner sentence like this, e-mail is perfectly fine.

Nejdřív means first, at first, or first of all.

It shows that this action happens before something else. In this sentence, it sets the sequence:

• Nejdřív musím napsat e-mail sekretářce.
• First I have to write an email to the secretary.

A close synonym is nejprve. Both are common.

Yes. Czech word order is flexible, although different orders can sound more natural in different contexts.

The given sentence is very natural:

• Nejdřív musím napsat e-mail sekretářce.

But you could also hear:

• Musím nejdřív napsat e-mail sekretářce.
• E-mail sekretářce musím nejdřív napsat.

These versions shift the emphasis a little:

• putting nejdřív first emphasizes the sequence
• putting musím earlier emphasizes the necessity
• putting e-mail sekretářce earlier emphasizes what has to be written

For a learner, the original sentence is a very good neutral model.

Because Czech usually does not need subject pronouns when the verb form already shows who is doing the action.

• musím already means I must

So já is unnecessary unless you want extra emphasis or contrast.

Compare:

• Musím napsat e-mail sekretářce. = normal, neutral
• Já musím napsat e-mail sekretářce. = I have to write the email (maybe not someone else)

Yes. Sekretářka is grammatically and semantically female.

So sekretářce means to a female secretary.

If the secretary were male, you would need a different noun, for example:

• sekretáři = to a male secretary
• or sometimes a different job title such as asistentovi = to an assistant

Czech often marks gender more explicitly than English does.

A rough English-style guide is:

• Nejdřív ≈ NAY-dzheef / NAY-drzheev
• musím ≈ MOO-seem
• napsat ≈ NUP-saht
• e-mail ≈ EH-mail
• sekretářce ≈ seh-kreh-TAARH-tseh

A few notes:

• ř in sekretářce is a special Czech sound and is difficult for learners
• á is a long vowel
• í is also long
• stress in Czech is usually on the first syllable of each word

So the stress pattern is roughly:

• NEJdřív
• MUsím
• NApsat
• E-mail
• SEkretářce

Yes, but it means something slightly different.

• napsat e-mail sekretářce = write an email to the secretary
• poslat e-mail sekretářce = send an email to the secretary

So napsat focuses on composing/writing the email, while poslat focuses on sending it.

You could even combine them:

• Nejdřív musím napsat e-mail a potom ho poslat sekretářce.
• First I have to write the email and then send it to the secretary.

In this sentence:

• e-mail = accusative singular
  It is the thing being written.
• sekretářce = dative singular
  It is the recipient.

So the structure is:

• musím napsat
  • what? → e-mail
• komu? (to whom?) → sekretářce

Those two questions, co? and komu?, are very useful for identifying the cases in sentences like this.