Вратарь после матча сказал, что первый гол был случайным.

Yes. Вратарь is the standard Russian word for goalkeeper.

A few useful notes:

• It comes from врата/ворота, meaning gate/goal.
• In sports Russian, you may also hear голкипер, which is a borrowing from English.
• Вратарь is the more traditional and fully native-sounding word.

So in this sentence, вратарь simply means the goalkeeper.

Because после requires the genitive case.

• матч = match (dictionary form, nominative)
• матча = genitive singular

So:

• после матча = after the match

This is a very common pattern in Russian:

• после урока = after the lesson
• после игры = after the game
• после фильма = after the film

Сказал is the past tense masculine singular form of сказать.

It matches the subject:

• вратарь is masculine singular
• so the verb is сказал

Compare:

• он сказал = he said
• она сказала = she said
• они сказали = they said

So the form is chosen because the speaker in the sentence is the goalkeeper, a masculine noun.

This is an aspect question.

• сказать is perfective
• говорить is imperfective

In this sentence, сказал is used because it refers to one completed act of speaking: he made a statement after the match.

So:

• сказал = said / made the statement
• говорил = was saying, used to say, or sometimes said with more focus on the process or repeated speaking

Here, сказал is the most natural choice because the sentence reports a single finished comment.

Что here means that and introduces a subordinate clause.

So the structure is:

• Вратарь ... сказал, что ...
• The goalkeeper ... said that ...

This is one of the most common ways to report speech in Russian.

For example:

• Он сказал, что устал. = He said that he was tired.
• Она сказала, что придёт. = She said that she would come.

Because что introduces a subordinate clause, and Russian normally puts a comma before subordinating conjunctions like что.

So:

• сказал, что...
• думал, что...
• знал, что...

This is standard Russian punctuation.

Because первый гол is the subject of the clause что первый гол был случайным.

Both words are in the nominative singular masculine:

• гол is a masculine noun
• первый agrees with гол in gender, number, and case

So:

• первый гол = the first goal

If the noun changed, the adjective would change too:

• первая игра = the first game
• первое место = first place
• первые минуты = the first minutes

Russian usually omits to be only in the present tense.

For example:

• Он врач. = He is a doctor.
• literally: He doctor

But in the past tense, Russian does use forms of быть:

• был = was (masculine singular)
• была = was (feminine singular)
• было = was (neuter singular)
• были = were

So:

• гол был случайным = the goal was accidental/by chance

The form был agrees with гол, which is masculine singular.

Because after был, Russian very often uses the instrumental case for a predicate noun or adjective describing what something was.

Here:

• случайный = nominative
• случайным = instrumental

So:

• гол был случайным = the goal was accidental / was a fluke

This instrumental pattern is very common and natural in past-tense sentences with быть.

A rough comparison:

• Он был врачом. = He was a doctor.
• Матч был интересным. = The match was interesting.
• Гол был случайным. = The goal was accidental.

You may sometimes see nominative predicates too, but in this sentence случайным is the normal choice.

Here случайным means something like:

• accidental
• by chance
• not really planned
• in sports English, possibly a fluke

It does not mean random in a mathematical or modern internet-style sense.

In football/soccer context, the speaker is saying the first goal was not the result of a deliberate, well-executed move, but happened more by chance.

Yes. Russian word order is fairly flexible.

This sentence:

• Вратарь после матча сказал, что первый гол был случайным.

could also be:

• После матча вратарь сказал, что первый гол был случайным.

Both are natural.

The difference is mainly in emphasis:

• Вратарь после матча сказал... puts the goalkeeper first
• После матча вратарь сказал... puts after the match first

The original order is a neutral, natural way to present the information.

The stress is:

• врата́рь
• по́сле
• ма́тча
• сказа́л
• что
• пе́рвый
• гол
• был
• случа́йным

So the full sentence with stress marks is:

Врата́рь по́сле ма́тча сказа́л, что пе́рвый гол был случа́йным.

A small note:

• что is often unstressed or only lightly stressed in normal speech.