После кофе мне трудно заснуть.

После means after, and it requires the genitive case.

So in principle:

• после + genitive

That is why кофе is in the genitive here.

The useful thing to know is that кофе is usually treated as an indeclinable noun, so its form often stays кофе in different cases. In other words, it is genitive here, but it looks the same as the dictionary form.

Examples:

• после урока = after the lesson
• после работы = after work
• после кофе = after coffee

Because кофе is one of those Russian nouns that usually do not change form.

So although the case changes grammatically, the word itself stays:

• nominative: кофе
• genitive: кофе
• dative: кофе
• etc.

This is common with some borrowed words in Russian.

Also, кофе is traditionally treated as masculine, although in modern everyday speech some speakers may treat it more loosely. In this sentence, that gender issue does not affect the form.

Because Russian often uses the dative case to show the person who experiences a feeling or condition.

Here, мне трудно заснуть literally means something like:

• to me, it is difficult to fall asleep

So:

• я = I
• мне = to me / for me

This pattern is very common in Russian:

• мне трудно = it is hard for me
• мне холодно = I am cold
• мне скучно = I am bored
• мне нужно уйти = I need to leave

English usually uses I, but Russian often uses dative + an impersonal expression.

Because трудно is not an adjective here. It is being used as a predicative adverb / category of state word, meaning it is difficult.

Compare:

• трудный экзамен = a difficult exam
  Here трудный is an adjective describing a noun.

• мне трудно заснуть = it is difficult for me to fall asleep
  Here трудно describes the situation, not a noun.

This is a very common Russian pattern:

• легко понять = it is easy to understand
• трудно сказать = it is hard to say
• невозможно сделать = it is impossible to do

So трудно is exactly the form you want before an infinitive like заснуть.

Because Russian often omits the present tense of “to be”.

So where English says:

• It is difficult for me to fall asleep

Russian simply says:

• мне трудно заснуть

There is no present-tense is in normal Russian sentences like this.

Also, this sentence is impersonal. It does not have a normal subject like it. Russian often expresses ideas like difficulty, necessity, comfort, temperature, and emotions without a subject:

• мне трудно = it is hard for me
• мне хорошо = I feel good / it is good for me
• мне надо идти = I need to go

Because заснуть means to fall asleep, while спать means to sleep.

That distinction matters here.

• заснуть = to begin sleeping, to fall asleep
• спать = to be asleep, to sleep

After coffee, the problem is usually that you cannot fall asleep, not that you cannot perform the state of sleeping in general. So:

• После кофе мне трудно заснуть = After coffee, it’s hard for me to fall asleep.

If you said трудно спать, it would sound more like it is hard to sleep, which can work in some contexts, but it is not quite the same idea.

Because заснуть focuses on the single completed change of state: the moment of falling asleep.

Russian aspect here works very naturally:

• засыпать = to be falling asleep / to fall asleep repeatedly or habitually / process-oriented imperfective
• заснуть = to fall asleep once, successfully, perfective

With трудно plus an infinitive, Russian often uses the aspect that best matches the intended meaning. Here, the meaning is about the difficulty of achieving the result of sleep onset, so заснуть is the normal choice.

You could compare:

• Мне трудно засыпать после кофе.
  = I have difficulty falling asleep after coffee.
  This sounds more general/habitual or process-focused.

• После кофе мне трудно заснуть.
  = After coffee, it’s hard for me to fall asleep.
  This sounds very natural and result-focused.

In normal usage, после кофе most naturally means after drinking coffee or after having coffee.

Russian often leaves that unstated because it is obvious from context. So it does not necessarily mean only after the cup is physically finished; it means after coffee has had its effect / after I’ve had coffee.

If needed, Russian can be more explicit:

• после чашки кофе = after a cup of coffee
• после того, как я выпью кофе = after I drink coffee

But in everyday speech, после кофе is completely natural.

Yes, absolutely. That is a very natural alternative.

The difference is mostly in nuance:

• После кофе мне трудно заснуть
  = After coffee, it’s hard for me to fall asleep.
  Slightly softer; focuses on difficulty.

• После кофе я не могу заснуть
  = After coffee, I can’t fall asleep.
  Stronger and more direct; focuses on inability.

Both are common. The first sounds a bit more descriptive, while the second sounds more categorical.

Yes, Russian word order is fairly flexible, and the order can change depending on what you want to emphasize.

The given order is very natural:

• После кофе мне трудно заснуть.

It starts with the context after coffee, then says for me, then the main idea it is hard to fall asleep.

Other possible orders:

• Мне трудно заснуть после кофе.
  Very natural too; a neutral way to say it.

• Заснуть после кофе мне трудно.
  More emphasis on falling asleep after coffee as the difficult thing.

• Мне после кофе трудно заснуть.
  Also natural; slightly different rhythm.

So the original sentence is not the only correct order, but it is a very normal one.

It is neutral and very natural in everyday Russian.

It is not especially formal, and it is not slangy either. You could say it in conversation, write it in a message, or use it in standard written Russian.

Other neutral alternatives include:

• Мне трудно заснуть после кофе.
• После кофе я плохо засыпаю.
• После кофе я не могу заснуть.

All of these are natural, with slightly different shades of meaning.