После сада я заметила дырку на рукаве и решила зашить её дома.

Because после always takes the genitive case.

• Dictionary form: сад
• Genitive singular: сада

So после сада means after the garden / after kindergarten, depending on context.

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после обеда = after lunch

Those are past tense feminine singular forms.

In Russian past tense, the verb agrees with the subject in gender and number:

• я заметил, я решил = if the speaker is male
• я заметила, я решила = if the speaker is female

So this sentence tells us that the speaker is female.

Because дырку is the accusative singular form of дырка.

Here дырка is the direct object of заметила: she noticed a hole.

For many feminine nouns ending in -а, the accusative singular changes to -у:

• дырка → дырку
• книга → книгу
• машина → машину

So:

• дырка = nominative, dictionary form
• дырку = accusative, used here as the object

Because на can mean either on(to) or on/at, and the case depends on whether there is motion or location.

Here the hole is located on the sleeve, so Russian uses the prepositional case:

• на рукаве = on the sleeve

If there were movement onto something, Russian would usually use the accusative:

• положить на стол = put onto the table
• дыра на рукаве = a hole on the sleeve

So рукаве is the prepositional singular of рукав.

Because зашить is a perfective verb and it means something like to sew up / mend by sewing, with a sense of a completed result.

That fits the idea here: she decided to fix the hole completely.

Compare:

• шить = to sew, to be sewing, to sew in general
• зашить = to sew up, to stitch closed, to mend completely

So решила зашить её means decided to sew it up / mend it, not just decided to do some sewing.

After verbs like решить (to decide), Russian normally uses an infinitive to say what someone decided to do.

Pattern:

• решил / решила + infinitive

Examples:

• решил уйти = decided to leave
• решила купить = decided to buy
• решила зашить = decided to sew up

So this works very much like English decided to do something.

Её refers to дырку.

Since дырка is feminine singular, the pronoun is also feminine singular:

• дырка → её

Here it is the direct object of зашить: to sew it up.

Russian could sometimes omit an object if it is very obvious from context, but in this sentence её sounds natural because it clearly points back to дырку. Without it, решила зашить дома would sound incomplete or would suggest something more general like decided to do some sewing at home.

Because дома is an adverb meaning at home.

• дома = at home
• в доме = in the house / inside the building

So in this sentence, дома means she planned to do it at home, not necessarily to emphasize being physically inside a particular house as a building.

This is a very common word:

• Я дома = I’m at home
• Останусь дома = I’ll stay home

Because Russian often omits a repeated subject when it is clearly the same person.

So:

• я заметила ... и решила ...

means:

• I noticed ... and decided ...

Russian does not need to repeat я here. You could say я заметила ... и я решила ..., but that would usually sound more emphatic than necessary.

Yes. Russian word order is quite flexible because the cases show the grammatical relationships.

This sentence could be rearranged in different ways, for example:

• Я после сада заметила дырку на рукаве и решила зашить её дома.
• Дырку на рукаве я заметила после сада и решила зашить её дома.

The basic meaning stays the same, but the emphasis changes:

• putting после сада first emphasizes the time
• putting дырку на рукаве first emphasizes what was noticed

The original order sounds natural and neutral.