У него усталый вид после работы.

У него literally means by him / at him, but in Russian this structure is very commonly used to express possession or a characteristic someone has.

So:

• У него усталый вид = literally At him [there is] a tired appearance
• Natural English: He looks tired / He has a tired look

Russian often avoids using a direct verb like to have in the way English does.

Also, него is the form of он used after a preposition like у.

In the present tense, Russian often leaves out the verb to be.

So instead of something like:

• He has a tired appearance

Russian can simply say:

• У него усталый вид

The meaning is understood without an extra verb. This is completely normal in Russian.

If you translated it word-for-word into English, it would sound incomplete, but in Russian it is a full sentence.

Here вид means appearance, look, or expression.

So усталый вид means:

• a tired look
• a tired appearance

In other contexts, вид can also mean things like:

• type
• kind
• view

But here it definitely means appearance/look.

Because усталый is an adjective describing the noun вид.

• усталый вид = a tired look

But устал is the short-form predicate meaning:

• is tired
• got tired

Compare:

• Он устал. = He is tired.
• У него усталый вид. = He has a tired look.

So:

• устал describes him
• усталый describes вид

Because it has to agree with вид.

Вид is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives:

• усталый вид

If the noun changed, the adjective would change too. For example:

• усталое лицо = a tired face
• усталая женщина = a tired woman

This is normal adjective agreement in Russian.

Because the preposition после always requires the genitive case.

So:

• dictionary form: работа
• after после: работы

That is why you get:

• после работы = after work

This is something you generally just have to learn with the preposition:

• после урока = after the lesson
• после фильма = after the film
• после работы = after work

Usually it means after work in the everyday English sense, especially in a sentence like this.

So:

• У него усталый вид после работы.
  = He looks tired after work.

Depending on context, it could also mean after the work/job/task was finished, but the most natural interpretation is simply after work.

They are very close in meaning, but not exactly identical in tone.

• У него усталый вид = He has a tired look / He looks tired
• Он выглядит усталым = He looks tired

The first one focuses on his appearance as a noun phrase: a tired look.
The second uses the verb выглядеть = to look / appear.

Both are natural.
У него усталый вид can sound a little more observational or descriptive.

After many prepositions, Russian third-person pronouns add н- at the beginning.

So you get:

• у него
• для неё
• с ними

instead of:

• у его
• для её
• с ими

So here, because the pronoun comes after the preposition у, Russian uses него.

Yes. Russian word order is flexible.

For example, you could also say:

• После работы у него усталый вид.

Both versions are correct, but the emphasis changes a little.

• У него усталый вид после работы.
  Focuses first on him / his appearance
• После работы у него усталый вид.
  Focuses first on after work

So the original sentence is natural, but not the only possible order.

Yes. A very literal translation would be:

• He has a tired appearance after work
• He has a tired look after work

But the most natural English translation is usually:

• He looks tired after work.

So when reading Russian, it is often helpful to understand both:

• the literal structure
• the natural English equivalent