После долгого дня я пойду спать пораньше.

It is in the genitive case because the preposition после always takes the genitive.

So:

• день → дня
• долгий → долгого

That is why you get после долгого дня, not после долгий день. This is a very common pattern in Russian: после + genitive = after ...

In Russian, долгий день is the normal way to say a long day in the sense of a day that lasted a long time or felt tiring.

Длинный usually refers more to physical length, so it works better with things like длинная улица or длинная верёвка. With day, долгий is much more natural.

So for the English idea after a long day, после долгого дня is the idiomatic choice.

Пойду is the future form of the perfective verb пойти.

Here it means something like I will go or I’ll head off. In Russian, пойти + infinitive is a very common way to express go and do something:

• пойду спать = I’ll go to sleep / I’ll go to bed
• пойду поем would be different, but пойду поесть = I’ll go eat

By contrast:

• иду means I am going / I’m on my way right now
• буду идти means I will be going/walking, which does not fit naturally here

So я пойду спать is the normal choice.

Because after пойти, Russian often uses an infinitive to show the purpose of going.

So пойти спать literally means to go in order to sleep, which English usually expresses as go to sleep or go to bed.

This pattern is very common:

• пойти гулять = to go for a walk
• пойти работать = to go work
• пойти отдыхать = to go rest

So спать stays in the infinitive because it is the action the speaker is going to do.

Not exactly, though in many situations they are close.

• пойду спать focuses on going off to bed / heading to sleep
• лягу спать focuses more on lying down to sleep

So пойду спать is a bit broader and very conversational. It can imply I’m done for today; I’m going to bed now.
Лягу спать sounds slightly more focused on the moment of physically lying down.

In this sentence, пойду спать sounds very natural.

Пораньше means a bit earlier, earlier than usual, or somewhat early.

It is related to раньше (earlier), but the prefix по- often makes it sound softer or more practical: not just earlier, but a little earlier than normal/than expected.

So:

• раньше = earlier
• пораньше = a bit earlier / nice and early / earlier than usual

In this sentence, пойду спать пораньше means the speaker plans to go to bed earlier than they normally would.

No, Russian word order is fairly flexible, but the given order is natural and neutral:

После долгого дня я пойду спать пораньше.

This order works well because:

• После долгого дня sets the scene first
• я пойду спать gives the main action
• пораньше adds the final detail

You could move things around for emphasis, but not every version sounds equally natural. For example:

• Я после долгого дня пойду спать пораньше — also possible
• Я пойду спать пораньше после долгого дня — possible, but the last phrase may sound like an afterthought

So the original word order is a good standard model.

Because После долгого дня is just a normal introductory prepositional phrase, and in Russian that usually does not require a comma.

So:

• После долгого дня я пойду спать пораньше. — normal
• После долгого дня, я пойду... — usually not correct in standard punctuation

A comma might appear only for special intonation or a very unusual stylistic reason, but in ordinary writing you should leave it out.

In standard pronunciation, adjective endings like -ого and -его are usually pronounced more like -ово and -ево.

So долгого is pronounced approximately like долгова.

The sentence stress is:

Пóсле дóлгого дня́ я пойду́ спать пора́ньше.

That pronunciation detail is very common in Russian adjectives, so it is worth getting used to early.