Мне нужно починить выключатель в коридоре.

Because Russian often expresses need with the pattern кому-то нужно + infinitive, literally to someone it is necessary to...

So:

• мне = to me / for me
• нужно = necessary
• починить = to fix

So Мне нужно починить... literally means It is necessary for me to fix..., which is the natural Russian way to say I need to fix...

That is why Russian uses the dative case here, not the nominative я.

Нужно means necessary / needed in this structure.

In Мне нужно починить выключатель, it works like I need to fix the switch.

A very literal breakdown is:

• Мне = to me
• нужно = it is necessary
• починить = to fix

So the whole phrase is essentially It is necessary for me to fix the switch.

Russian often uses this impersonal construction instead of a direct equivalent of English I need.

Because after нужно, Russian normally uses an infinitive to say what action is necessary.

So:

• Мне нужно починить = I need to fix
• Мне нужно идти = I need to go
• Мне нужно работать = I need to work

This is very similar to English need + to + verb, except Russian simply uses нужно + infinitive.

Починить is the perfective form, while чинить is imperfective.

Here, починить is used because the speaker means fix it successfully / get it repaired as a completed task, not just be fixing / do repair work in general.

So:

• починить = to fix, to repair completely
• чинить = to repair, to be repairing, to do repair work

In this sentence, the idea is probably I need to get the switch fixed, so the completed-result idea makes починить the natural choice.

Выключатель is the direct object of починить, so it is in the accusative case.

However, for many inanimate masculine nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: выключатель
• accusative: выключатель

That is why there is no visible ending change here.

Выключатель usually means switch, especially something like a light switch or electrical switch.

In this sentence, because of в коридоре (in the hallway/corridor), the most natural interpretation is the light switch in the hallway.

It comes from the verb выключать / выключить, meaning to switch off / turn off.

Because в коридоре expresses location: in the corridor / hallway.

With в, Russian uses different cases depending on meaning:

• в + accusative = motion into somewhere
  
  • в коридор = into the corridor
• в + prepositional = location in somewhere
  
  • в коридоре = in the corridor

Here the switch is located in the hallway, so Russian uses в коридоре.

In this sentence, the most natural meaning is that the switch is in the corridor:

Мне нужно починить выключатель в коридоре.
= I need to fix the switch in the corridor.

Grammatically, Russian often leaves this kind of attachment to context. In theory, someone could imagine fix the switch while in the corridor, but that is much less likely. Native speakers would normally understand в коридоре as describing выключатель here.

If you wanted to make it even clearer, context or intonation would usually do that.

No, Russian word order is quite flexible.

The neutral order here is:

Мне нужно починить выключатель в коридоре.

But other orders are possible depending on emphasis, for example:

• Выключатель в коридоре мне нужно починить.
  Emphasizes the switch in the corridor
• Починить выключатель в коридоре мне нужно.
  Sounds more marked, often emphasizing the action or sounding more stylistic

Even though word order can change, the original sentence is the most natural neutral version.

Yes, but it is not exactly the same.

• Мне нужно починить... = I need to fix... / it is necessary for me to fix...
• Я должен починить... = I must / am supposed to fix...

Должен often sounds stronger, more like an obligation, duty, or expectation.
Нужно is often softer and more neutral.

So if you simply mean I need to fix the switch, мне нужно is usually the better match.

Yes. Мне надо починить выключатель в коридоре is also natural and means almost the same thing.

In many everyday contexts, надо and нужно are very close:

• мне надо...
• мне нужно...

Very roughly:

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

But in this sentence, both work well.

A simple pronunciation guide:

• Мне = mnye
• нужно = NOOZH-na
• починить = pa-chee-NEET'
• выключатель = vy-klyu-CHA-tyel'
• в коридоре = f ka-ri-DO-rye

A fuller approximation:

mnye NOOZH-na pa-chee-NEET' vy-klyu-CHA-tyel' f ka-ri-DO-rye

A couple of useful points:

• в before к is pronounced like f
• the stress matters a lot:
  • нужно
  • почини́ть
  • выключа́тель
  • коридо́ре

Not in this sentence.

Here, нужно is being used in an impersonal construction, so it stays the same:

• Мне нужно идти
• Ему нужно работать
• Им нужно ждать

It does not agree with мне or with выключатель.

So even though выключатель is masculine singular, you still say нужно, not something else.