Poslije napornog dana treba mi mir.

Poslije means after.

Yes, you could also say nakon: Nakon napornog dana treba mi mir.

Both are correct and both take the genitive case after them.
A small difference in feel:

• poslije = very common and everyday
• nakon = also common, sometimes a bit more formal or neutral

Because poslije requires the genitive case.

So:

• dan = day → genitive singular dana
• naporan = tiring / hard / exhausting → genitive singular masculine napornog

So poslije napornog dana literally means after of a tiring day, but in natural English just after a tiring day.

In Croatian, adjectives must agree with the noun they describe in:

• gender
• number
• case

Here the noun is dan, which becomes dana because it is in the genitive singular after poslije.
So the adjective must also be genitive singular masculine:

• nominative: naporan dan
• genitive: napornog dana

This is a very common Croatian way to say I need something.

Croatian often uses trebati like this:

• treba mi mir = I need peace/quiet
• literally: peace is needed to me

So:

• treba = is needed
• mi = to me

This structure is very natural and standard in Croatian.

You may sometimes hear trebam mir in speech, but treba mi mir is the safest and most standard choice.

Mi is the dative singular clitic form of ja (I).

It means to me.

Compare:

• mi = short/clitic form
• meni = full/emphatic form

So:

• Treba mi mir. = I need peace/quiet.
• Meni treba mir. = I need peace/quiet with extra emphasis on me

Mir is in the nominative singular.

That may feel strange to an English speaker, because in English peace looks like the object of need. But in this Croatian structure, the thing needed acts more like the grammatical subject.

That is why the verb agrees with it:

• Treba mi mir. = singular noun → treba
• Trebaju mi dva dana odmora. = plural noun phrase → trebaju

So mir is not in the accusative here.

It is singular because mir is singular.

In this pattern, the verb agrees with the thing that is needed:

• Treba mi mir. = peace is singular
• Trebaju mi odgovori. = answers are plural

So the verb is not agreeing with mi. It is agreeing with mir.

Croatian word order is fairly flexible, and this sentence has a natural, neutral order.

Poslije napornog dana treba mi mir. sounds normal.

You can also hear other orders, depending on emphasis, for example:

• Poslije napornog dana mi treba mir.
• Poslije napornog dana mir mi treba. → stronger emphasis on mir
• Poslije napornog dana meni treba mir. → stronger emphasis on meni

So the meaning stays similar, but the focus changes slightly.

It can mean both.

The basic meaning of mir is peace, but in everyday sentences like this it often means:

• peace and quiet
• calm
• being left alone

So here treba mi mir is often understood as something like:

• I need some peace
• I need some quiet
• I need some calm

Croatian often leaves out subject pronouns when they are not needed.

English needs I need..., but Croatian does not have to say ja here, because mi already shows who the sentence is about.

So:

• Treba mi mir. = natural
• Ja trebam mir. = possible in some speech, but not the best standard model here
• Meni treba mir. = natural if you want emphasis on me

Because Croatian has no articles.

So dan can mean:

• a day
• the day

and mir can mean:

• peace
• the peace
• sometimes even some peace, depending on context

The exact meaning is understood from the situation, not from an article.

In standard Croatian, poslije is the normal form.

Posle is associated mainly with Serbian. Since your sentence is Croatian, poslije is the expected choice.

So for Croatian, it is best to learn:

• poslije = after

and recognize posle when you hear it in other varieties.