Нужно перешить подкладку.

Because this is an impersonal construction.

In Russian, нужно + infinitive means something like:

• it is necessary to...
• one needs to...
• ...needs to be done

So the sentence does not have to say who exactly needs to do it.

• Нужно перешить подкладку. = The lining needs to be resewn / It’s necessary to resew the lining.

If you want to say who needs to do it, Russian often adds that person in the dative case:

• Мне нужно перешить подкладку. = I need to resew / have the lining resewn.
• Нам нужно перешить подкладку. = We need to resew / have the lining resewn.

Нужно means necessary, needed, or must be done.

In this sentence, it works like a predicative word: it expresses necessity, not an action.

So:

• нужно перешить = it is necessary to resew
• нужно сделать = it is necessary to do
• нужно купить = it is necessary to buy

A useful way to think of it is:

• нужно + infinitive = need to + verb

This нужно does not change for person or number here.

Because нужно is followed by an infinitive.

Russian commonly uses this pattern:

• нужно + infinitive
• надо + infinitive
• можно + infinitive
• нельзя + infinitive

Examples:

• Нужно подождать. = It’s necessary to wait.
• Можно войти. = One may enter / You can come in.
• Нельзя курить. = Smoking is not allowed.

So перешить stays in the dictionary form because it names the action that is necessary.

The base verb is шить = to sew.

With the prefix пере-, перешить usually means:

• to resew
• to sew again
• sometimes to alter by sewing

So перешить подкладку means something like:

• to resew the lining
• to redo the lining
• depending on context, to replace/alter the lining by sewing

The prefix пере- often gives the sense of again, over, or redoing something.

Перешить is perfective.

That means it focuses on the action as a completed result:

• перешить = to resew and finish it
• imperfective: перешивать = to be resewing / to resew repeatedly / to work on resewing

After нужно, both aspects are possible in Russian, but the choice changes the meaning.

Here, перешить is natural because the speaker usually means:

• the job needs to be done
• the lining needs to end up resewn

So the focus is on the result, not the process.

Because it is the direct object of the transitive verb перешить.

The noun is:

• nominative: подкладка = lining
• accusative: подкладку

Since подкладка is a feminine noun ending in -а, its accusative singular changes to -у:

• подкладка → подкладку

So:

• перешить что? → подкладку

That is exactly the question Russian grammar asks for the direct object: what needs to be resewn?

Подкладка means lining, especially the inner fabric layer of something like:

• a coat
• jacket
• dress
• skirt
• bag

So in this sentence, it most likely refers to the lining of a piece of clothing or another sewn item.

Yes. Russian word order is flexible.

This sentence can also be said as:

• Подкладку нужно перешить.

Both versions are grammatical.

The difference is mostly about focus:

• Нужно перешить подкладку.
  Focuses first on the fact that something needs to be done.

• Подкладку нужно перешить.
  Brings the lining to the front, often contrasting it with something else.

For example:

• Рукава можно оставить, а подкладку нужно перешить.
  = The sleeves can stay, but the lining needs to be resewn.

Yes. Надо and нужно are often close in meaning.

• Нужно перешить подкладку.
• Надо перешить подкладку.

Both can mean The lining needs to be resewn.

Very roughly:

• надо can sound a bit more conversational and direct
• нужно can sound slightly more neutral or formal

But in many everyday contexts, they are interchangeable.

By itself, the Russian sentence does not make that clear.

Нужно перешить подкладку only tells you that the action is necessary. It does not specify whether:

• you will do it yourself
• someone else will do it
• you will take it to a tailor

Russian often leaves that unspecified unless context makes it clear.

For example:

• Мне нужно перешить подкладку.
  Could mean I need to resew the lining or I need to get the lining resewn, depending on context.

The stress is:

• ну́жно
• переши́ть
• подкла́дку

So the full sentence is pronounced approximately:

• Ну́жно переши́ть подкла́дку.

For learners, the most important thing is not to move the stress in перешить or подкладку, because Russian stress is unpredictable and can affect naturalness a lot.