Da meine Schwester heute keine Zeit hat, schreibt sie mir nur eine kurze Nachricht.

Here da means since or because. It introduces a subordinate clause:

• Da meine Schwester heute keine Zeit hat = Since my sister doesn’t have time today

In everyday German, weil is also common for because, but da often sounds a bit more formal or is used when the reason is already known or obvious from context.

Because da introduces a subordinate clause, and in German subordinate clauses the conjugated verb goes to the end.

So:

• meine Schwester hat heute keine Zeit = normal main clause order
• da meine Schwester heute keine Zeit hat = subordinate clause, so hat moves to the end

This is one of the most important German word order rules.

In German, a subordinate clause is normally separated from the main clause by a comma.

So:

• Da meine Schwester heute keine Zeit hat, = subordinate clause
• schreibt sie mir nur eine kurze Nachricht. = main clause

This comma is required.

German main clauses follow the verb-second rule: the conjugated verb must be the second element.

Since the sentence starts with the subordinate clause Da meine Schwester heute keine Zeit hat, that whole clause counts as the first element. So the verb of the main clause must come next:

1. Da meine Schwester heute keine Zeit hat
2. schreibt
3. sie mir nur eine kurze Nachricht

That is why you get schreibt sie, not sie schreibt.

German usually uses kein to negate a noun when that noun has no article or could be thought of as having an indefinite article.

• Zeit is a noun
• So German says keine Zeit = no time

Compare:

• Ich habe keine Zeit.
• not: Ich habe nicht Zeit.

Use nicht mainly to negate verbs, adjectives, adverbs, or more specific parts of a sentence.

Meine Schwester is nominative, because she is the subject of hat in the subordinate clause.

You can think of it like this:

• Who has no time today? → meine Schwester

That makes it the subject, so nominative is used.

Because the verb schreiben often takes:

• a dative person = the recipient
• an accusative thing = what is written

So in this sentence:

• mir = to me → dative
• eine kurze Nachricht = the thing being written → accusative

Compare:

• Sie schreibt mir eine Nachricht.
• Mir is the person receiving the message.

This is an accusative noun phrase:

• eine = feminine accusative singular article
• kurze = adjective ending that matches the article and noun
• Nachricht = feminine noun

Because Nachricht is what she writes, it is the direct object, so accusative is used.

Since Nachricht is feminine singular, the accusative article is eine.

German adjectives usually take an ending when they come before a noun.

Here:

• eine is the article
• Nachricht is feminine singular
• so the adjective gets the ending -e

That gives:

• eine kurze Nachricht

This is part of German adjective declension. After ein/eine words, the adjective ending depends on gender, case, and number.

Heute is a time expression. In this sentence it appears in the subordinate clause:

• Da meine Schwester heute keine Zeit hat

Its position is natural and common. German word order with adverbs is flexible, but this placement sounds very normal.

You could also hear variations depending on emphasis, but meine Schwester heute keine Zeit hat is a standard arrangement.

Nur means only. It limits what follows:

• nur eine kurze Nachricht = only a short message

The idea is that because she has no time, she will write just that, and nothing more substantial.

Its position before eine kurze Nachricht is the normal one here.

Yes. You could say:

• Weil meine Schwester heute keine Zeit hat, schreibt sie mir nur eine kurze Nachricht.

That is grammatically correct and means the same thing in most contexts.

A common learner point:

• da can sound slightly more formal or written
• weil is extremely common in speech and everyday German

Both take subordinate clause word order, so the verb still goes to the end.

Yes. German often allows you to switch the order of the clauses:

• Meine Schwester schreibt mir heute nur eine kurze Nachricht, da sie keine Zeit hat.

Or:

• Sie schreibt mir nur eine kurze Nachricht, weil meine Schwester heute keine Zeit hat.

When the subordinate clause comes first, the main clause must begin with the verb because of the verb-second rule:

• Da ..., schreibt sie ...

When the main clause comes first, normal main clause order appears first:

• Sie schreibt ... , da ...

Because German often uses the dative case instead of a preposition where English uses to.

So:

• mir already means to me in this context

German does not need something like zu mir here. In fact, schreibt sie zu mir eine Nachricht would be wrong in this meaning.

Yes, very much so. Zeit haben is the normal German way to say to have time.

Examples:

• Ich habe keine Zeit.
• Hast du heute Zeit?
• Wir haben morgen Zeit.

So in your sentence, keine Zeit hat is a very natural, everyday expression.