Опытный мастер быстро починил старый стул.

Because починил is a past tense verb in Russian.

The infinitive is починить = to repair / to fix.
In the past tense, Russian often adds -л to the stem:

• починил = he repaired
• починила = she repaired
• починило = it repaired
• починили = they repaired

So починил tells you that the subject is masculine singular.

This is a question of aspect, which is very important in Russian.

• чинить = to be repairing / to repair (imperfective)
• починить = to repair successfully / to fix (perfective)

So:

• Он чинил стул = He was repairing the chair / He repaired the chair (focus on the process, or just the action in a general sense)
• Он починил стул = He repaired the chair / He fixed the chair (focus on the completed result)

In your sentence, починил shows that the repair was completed.

Because мастер is in the nominative case, which is the normal case for the subject.

In Опытный мастер быстро починил старый стул:

• опытный мастер = the person doing the action
• старый стул = the thing receiving the action

Russian often marks roles through case endings, not only through word order.

It is in the accusative case, but for masculine singular inanimate nouns, the accusative looks exactly like the nominative.

So:

• nominative: старый стул
• accusative: старый стул

They are the same in form because стул is:

• masculine
• singular
• inanimate

Compare that with an animate noun:

• Я вижу старого мастера = I see the old craftsman

Here the accusative changes because мастер is animate.

Because both adjectives agree with their nouns in gender, number, and case.

In this sentence:

• опытный describes мастер
• старый describes стул

Both nouns are:

• masculine
• singular

And here they appear in forms that use -ый.

So the adjective endings match the nouns they describe.

Опытный means experienced.
Мастер can mean master, craftsman, repairman, or skilled worker, depending on context.

So опытный мастер means something like:

• an experienced craftsman
• a skilled repairman
• an experienced worker

It does not necessarily mean master in the sense of being the boss or owner. Here it usually suggests someone skilled at practical work.

Russian often omits subject pronouns when they are not needed.

In English, you usually need he:

• He quickly repaired the old chair

In Russian, the verb form already gives useful information:

• починил tells you the subject is masculine singular

So saying Он is possible, but not necessary here:

• Он быстро починил старый стул = He quickly repaired the old chair

Without он, the sentence still sounds natural.

Yes, Russian word order is more flexible than English word order.

The neutral order here is:

• Опытный мастер быстро починил старый стул.

But you could also say:

• Опытный мастер починил старый стул быстро.
• Старый стул быстро починил опытный мастер.

These versions may sound more marked or emphasize different parts of the sentence.

In the original sentence, быстро before the verb is a very natural placement and simply means quickly.

It is an adverb.

• быстрый = fast, quick (adjective)
• быстро = quickly, fast (adverb)

Here it describes how the master repaired the chair:

• быстро починил = repaired quickly

Because старый стул is a noun phrase, not a full sentence.

It consists of:

• старый = old
• стул = chair

Together: old chair

If you wanted to say The chair is old, Russian would need a different structure, usually:

• Стул старый.

So in your sentence, старый стул is the object being repaired.

It can be used for many things that can be fixed or repaired.

For example:

• починить стул = fix a chair
• починить машину = fix a car
• починить часы = repair a watch
• починить компьютер = fix a computer

So it is a very common general verb for repairing something.

A simple pronunciation guide is:

Opytnyy master bystro pochinil staryy stul

Very roughly for an English speaker:

• Опытный ≈ uh-PIT-nyy
• мастер ≈ MAS-ter
• быстро ≈ BIS-tra
• починил ≈ puh-chee-NEEL
• старый ≈ STAR-yy
• стул ≈ stool

A few notes:

• Russian ы is a sound that does not exist exactly in English.
• ч is like ch in chair.
• у sounds like oo in boot.

A more natural rhythm would stress:

опЫтный мАстер бЫстро починИл стАрый стул