Перед выходом я посмотрела в зеркало, поправила блузку и надела кепку.

Because the preposition перед normally takes the instrumental case when it means before or in front of.

• выход = exit / going out / departure
• выходом = instrumental singular of выход

So:

• перед выходом = before leaving / before going out

Literally, it is something like before the going-out.

This is a very common Russian pattern:

• перед сном = before sleep / before going to bed
• перед работой = before work
• перед отъездом = before departure

So перед выходом is just the normal case form required by перед.

Here выход refers to the act of going out / leaving.

Depending on context, выход can mean:

• exit as a physical way out
• way out / solution
• departure / going out

In this sentence, перед выходом means:

• before going out
• before leaving

So it does not mean the speaker is standing physically in front of an exit. It means this happened just before she left.

The -а ending shows that the subject is feminine in the past tense.

In Russian past tense, verbs agree with gender and number:

• посмотрел = he looked
• посмотрела = she looked
• посмотрело = it looked
• посмотрели = they looked

So:

• я посмотрела
• поправила
• надела

all tell you that the speaker or female subject is a woman/girl.

If the speaker were male, it would be:

• Перед выходом я посмотрел в зеркало, поправил блузку и надел кепку.

Because the sentence describes a sequence of completed actions.

Russian often uses the perfective aspect for single, finished actions in a story:

• посмотрела = looked
• поправила = adjusted / straightened
• надела = put on

The idea is:

1. she looked in the mirror,
2. adjusted her blouse,
3. put on her cap,

and then presumably left.

If you used imperfective forms, it would sound more like process, repetition, or background action:

• смотрела
• поправляла
• надевала

Those would not fit as naturally for this neat list of one-time completed actions.

Because in Russian, when you mean look in the mirror (to see your reflection), the normal expression is:

• смотреть в зеркало
• посмотреть в зеркало

This is just the standard idiom.

Compare:

• смотреть в зеркало = to look in the mirror
• смотреть на зеркало = to look at the mirror itself as an object

So in your sentence, the meaning is clearly she looked in the mirror, not she looked at the mirror.

This is a very common learner question.

Even though в + accusative often suggests motion into something, Russian also uses смотреть в + accusative as a fixed pattern meaning to look into something:

• смотреть в окно = look out the window / look into the window area
• смотреть в небо = look at the sky
• смотреть в зеркало = look in the mirror

So this is not really about physical movement. It is about the direction of the gaze.

That is why зеркало appears as:

• nominative: зеркало
• accusative singular: зеркало

For neuter inanimate nouns, nominative and accusative look the same, so the form does not change.

Here поправила блузку means something like:

• adjusted her blouse
• straightened her blouse
• fixed her blouse a little

The verb поправить often means to set right, to straighten, to adjust, or to fix.

Examples:

• поправить волосы = fix/straighten one’s hair
• поправить воротник = straighten one’s collar
• поправить одежду = adjust one’s clothes

So she probably checked herself in the mirror and then made a small adjustment to how the blouse was sitting.

Because both nouns are direct objects, so they are in the accusative case.

Base forms:

• блузка = blouse
• кепка = cap

Accusative singular for most feminine nouns ending in -а changes to -у:

• блузка → блузку
• кепка → кепку

That is why you get:

• поправила блузку
• надела кепку

This is a very common pattern:

• читать книгу
• купить ручку
• открыть дверь

This is a classic Russian learner issue.

The standard distinction is:

• надеть = to put on something
• одеть = to dress someone

So:

• надеть кепку = put on a cap
• надеть куртку = put on a jacket

But:

• одеть ребёнка = dress a child
• одеть куклу = dress a doll

A common memory trick in Russian is:

• Надеть что-то, одеть кого-то
  = Put on something, dress someone

So надела кепку is the correct standard choice here.

Because Russian does not need to repeat the subject when it stays the same.

So this is completely natural:

• Перед выходом я посмотрела в зеркало, поправила блузку и надела кепку.

The subject я applies to all three verbs:

• посмотрела
• поправила
• надела

Repeating я would usually sound unnecessary:

• я посмотрела..., я поправила..., я надела...

That might be used only for emphasis, contrast, or a special rhythm.

Yes, it could be rearranged, but the original word order is very natural.

Original:

• Перед выходом я посмотрела в зеркало, поправила блузку и надела кепку.

This sounds like a smooth narrative sequence: first the time frame, then the subject, then the actions.

Russian word order is flexible, but changes can affect emphasis. For example:

• Я перед выходом посмотрела в зеркало, поправила блузку и надела кепку.
  Also fine; slightly more emphasis on I.

• В зеркало я посмотрела перед выходом...
  This puts special focus on the mirror or on contrast.

So the original order is probably the most neutral and natural for ordinary narration.

Yes, that is a very good natural translation.

A few possible English versions are:

• Before going out, I looked in the mirror, adjusted my blouse, and put on a cap.
• Before leaving, I looked at myself in the mirror, straightened my blouse, and put on my cap.

The Russian sentence itself does not explicitly say my blouse or my cap, but English often adds my because it sounds natural when talking about your own clothes.

So in translation, adding my is often perfectly appropriate even though Russian does not use a possessive here.