Хотя брат и устал после работы, он всё равно успел побриться, надеть чистую рубашку и прийти вовремя.

Yes, хотя by itself already means although / though.

In sentences like this, и is often added after the subject or another important word for emphasis:

Хотя брат и устал..., он всё равно...

This pattern is very common in Russian and means something like:

• Although my brother was tired...
• Even though my brother was tired...

The и does not mean normal and here. It adds a concessive, emphatic feel: despite that fact.

You will often see this pattern:

• Хотя он и молодой, он очень опытный.
• Хотя было и поздно, они продолжали работать.

So хотя ... и ... is a useful structure to recognize.

Устал is the past tense of устать, which literally means to get tired. But in many contexts, Russian uses this form where English would naturally say was tired.

So:

• брат устал = the brother got tired / was tired

In this sentence, it describes the resulting state after work: he ended up tired after work.

Russian often prefers this short verbal form instead of an adjective phrase:

• Я устал. = I’m tired / I got tired.
• Она устала после поездки. = She was tired after the trip.

If you said был усталый, that would sound less natural here. It focuses more on being a tired kind of person / being in a tired condition, while устал is the standard everyday way to say was tired.

Because после requires the genitive case.

The base form is работа. In the genitive singular, it becomes работы.

So:

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

This is something worth memorizing:
после + genitive

Всё равно means something like:

• all the same
• nevertheless
• still
• anyway

In this sentence, it shows contrast:

• he was tired,
• but despite that, he still managed to do those things.

So:

он всё равно успел...
= he still / nevertheless managed to...

This expression is extremely common in Russian.

Examples:

• Я устал, но всё равно пойду. = I’m tired, but I’ll go anyway.
• Было холодно, но они всё равно гуляли. = It was cold, but they still went for a walk.

Успел comes from успеть, which often means:

• to manage to
• to have time to
• to succeed in doing something before it’s too late

So он успел побриться... means not just that time existed in theory, but that he did manage to shave, put on a clean shirt, and arrive on time.

It often implies a deadline or limited time.

Compare:

• Я успел поесть. = I managed to eat / I had time to eat.
• Я не успел позвонить. = I didn’t manage to call in time.

Here, успел is perfective past tense, which fits because the action was successfully completed.

Because they depend on успел.

In Russian, after verbs like успеть, хотеть, мочь, любить, and many others, the next action is usually expressed with an infinitive.

So:

• успел побриться = managed to shave
• успел надеть рубашку = managed to put on a shirt
• успел прийти вовремя = managed to arrive on time

This is very similar to English:

• managed to shave
• wanted to go
• can do

So the structure is normal:

успеть + infinitive

Because побриться is perfective, while бриться is imperfective.

Here the sentence is about a completed result: he managed to finish shaving.

• бриться = to be shaving / to shave regularly / to shave in general
• побриться = to shave oneself once, successfully, to completion

Since the sentence lists completed actions that he managed to do before arriving, perfective verbs are natural:

• побриться
• надеть
• прийти

That gives the sense of completed steps.

The -ся marks it as a reflexive verb.

Брить means to shave something or someone.
Бриться / побриться means to shave oneself.

So:

• брить бороду = to shave a beard
• бриться = to shave oneself

In English, we usually just say to shave, but in Russian the reflexive ending is often needed to show that the person is doing it to themselves.

This is a classic Russian distinction.

• надеть = to put on an item of clothing
• одеть = to dress a person

So:

• надеть рубашку = to put on a shirt
• одеть ребёнка = to dress a child

Because the direct object here is рубашку (the clothing item), надеть is the correct verb.

A common memory trick is:

• надеть что? — an item
• одеть кого? — a person

So надеть чистую рубашку is exactly right.

It is accusative singular feminine.

The verb надеть takes a direct object:
надеть что? = put on what?

The noun is:

• nominative: чистая рубашка
• accusative: чистую рубашку

Because рубашка is a feminine noun, both the adjective and noun change in the accusative singular:

• чистая → чистую
• рубашка → рубашку

So the phrase means a clean shirt as the object of put on.

Because the sentence refers to one completed arrival, not a habitual action.

• приходить = imperfective, to come / to arrive in general, repeatedly, or as a process
• прийти = perfective, to arrive once, successfully

Here he managed to arrive on time on one specific occasion, so прийти is the natural choice.

Compare:

• Он всегда приходит вовремя. = He always arrives on time.
  (habitual, imperfective)

• Он успел прийти вовремя. = He managed to arrive on time.
  (one completed event, perfective)

Because вовремя is a fixed adverb meaning:

• on time
• at the right time

It is normally written as one word.

Examples:

• Он пришёл вовремя. = He arrived on time.
• Нужно закончить вовремя. = You need to finish on time.

Do not confuse it with expressions involving во время in other contexts, where the meaning can be more like during in certain set phrases. In this sentence, вовремя is definitely the adverb on time.

Because Хотя брат и устал после работы is a subordinate clause introduced by хотя.

Russian separates that clause from the main clause with a comma:

Хотя брат и устал после работы, он всё равно успел...

This is similar to English punctuation with an introductory although clause:

• Although he was tired after work, he still managed...

So the comma marks the boundary between:

1. the concessive clause: Хотя брат и устал после работы
2. the main clause: он всё равно успел...

Because once the subject is established, Russian does not need to repeat it if it is clear.

In:

он всё равно успел побриться, надеть чистую рубашку и прийти вовремя

all three infinitives depend on успел, and the understood subject of all of them is он.

So it means:

• he managed to shave,
• he managed to put on a clean shirt,
• he managed to arrive on time.

This works just like English:
He managed to shave, put on a clean shirt, and arrive on time.

You do not need to repeat he before every verb.

The word order is fairly natural, but Russian does allow some flexibility.

The given sentence is standard and smooth:

Хотя брат и устал после работы, он всё равно успел...

This order presents:

1. the concession first: Although he was tired after work
2. the main point second: he still managed...

You could rearrange some parts for emphasis, but the original is probably the most neutral and natural for a learner to imitate.

For example, всё равно can move a little, but not freely in every position. The sentence as given is a very good model.

So yes, word order is somewhat flexible in Russian, but this version is idiomatic and safe to learn first.