Перед выходом мне нужно погладить рубашку.

Because the preposition перед requires the instrumental case when it means before or in front of something.

So:

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

That is why Russian says:

• перед выходом = before going out / before leaving

This is a very common pattern:

• перед уроком = before the lesson
• перед работой = before work
• перед сном = before sleep

Literally, it means before the exit / before going out, but in natural English it usually means before leaving or before going out.

In this sentence, выход is being used as a noun rather than a verb. Russian often uses noun phrases like this where English would prefer a verb:

• Перед выходом мне нужно... = Before leaving, I need to...
• More literally: Before going out, to me it is necessary...

So even though the noun is выход (exit / going out), the idea is clearly before I leave.

Because Russian uses an impersonal construction here.

• мне is the dative case of я
• нужно means necessary / needful

So мне нужно literally means something like:

• to me, it is necessary
• or more naturally, I need to

This is a very common Russian structure:

• мне нужно работать = I need to work
• тебе нужно идти = you need to go
• нам нужно купить хлеб = we need to buy bread

So Russian does not say я нужно here.

All three can express necessity, but they differ a little in tone.

• нужно = need to / necessary
  Neutral and very common.
• надо = need to / have to
  Also very common, often a bit more conversational.
• необходимо = necessary / it is necessary
  More formal or bookish.

In this sentence:

• Мне нужно погладить рубашку. = neutral, everyday
• Мне надо погладить рубашку. = also very natural
• Мне необходимо погладить рубашку. = more formal or emphatic

For everyday speech, нужно and надо are the most useful.

Because погладить is perfective, while гладить is imperfective.

Here the speaker means a single complete action:

• I need to iron the shirt
  meaning: I need to get that task done

That is why погладить is used.

Compare:

• Мне нужно погладить рубашку. = I need to iron the shirt.
  Focus on completing the task.
• Я глажу рубашку. = I am ironing the shirt.
  Focus on the process.
• Я часто глажу рубашки. = I often iron shirts.
  Repeated/habitual action.

So after нужно, both aspects are possible in general, but the perfective often sounds better when you mean one concrete thing that needs to be done.

Yes, погладить can mean both:

• to stroke / pet
• to iron

The meaning depends on the object.

Examples:

• погладить кошку = to pet the cat
• погладить рубашку = to iron the shirt

So in this sentence, because the object is рубашку (shirt), the meaning is clearly to iron.

Because рубашку is the accusative case, and it is the direct object of the verb погладить.

• рубашка = nominative singular
• рубашку = accusative singular

Since the action is being done to the shirt, Russian uses the accusative:

• погладить рубашку = to iron the shirt

This is very normal for transitive verbs:

• читать книгу = to read a book
• купить хлеб = to buy bread
• закрыть дверь = to close the door

Because Russian often leaves out possessive words like my, your, or his when the meaning is obvious from context.

So:

• Мне нужно погладить рубашку. naturally suggests I need to iron my shirt
• But literally it just says I need to iron the shirt

If needed, Russian can make it explicit:

• Мне нужно погладить свою рубашку. = I need to iron my shirt

However, in many everyday situations, свою is unnecessary because listeners will automatically understand whose shirt it is.

Yes. Russian word order is flexible.

The original sentence:

• Перед выходом мне нужно погладить рубашку.

A very natural alternative is:

• Мне нужно погладить рубашку перед выходом.

Both mean the same basic thing: Before leaving, I need to iron the shirt.

The difference is mostly about emphasis:

• Перед выходом... puts the time frame first: Before leaving...
• Мне нужно... starts with the need itself: I need to...

Russian often moves parts of the sentence around to highlight what is most important in the context.

Yes, absolutely.

That version means:

• Before going out / before I leave, I need to iron the shirt

Compare the two:

• Перед выходом мне нужно погладить рубашку.
  More compact; uses a noun phrase.
• Перед тем как выйти, мне нужно погладить рубашку.
  More explicit; uses a clause with a verb.

Both are correct. The first one is shorter and very natural; the second may feel a bit clearer to learners because it directly contains the idea to go out.

Yes, other nouns are possible, depending on the nuance.

For example:

• Перед выходом = before going out / before leaving
• Перед уходом = before leaving / before departure

These are similar, but not always identical in feel:

• выход emphasizes going out
• уход emphasizes leaving / departure

In many everyday contexts, both could work. In this sentence, перед выходом sounds natural if the person means before going out somewhere.

A few natural conversational versions are:

• Мне надо погладить рубашку перед выходом.
• Перед выходом надо погладить рубашку.
• Мне ещё нужно погладить рубашку перед выходом.
  (I still need to iron the shirt before leaving.)

The original sentence is already completely natural. These are just slight variations in tone or emphasis.