kousikisaito de sirabetara, hakubutukan ga yasumi ka douka sugu wakarimasita.

で marks the place or means where an action happens.

Here, 公式サイトで調べたら means when/after I checked on the official website.

So で is not marking destination like に would. It is showing the location or medium used for the action 調べる.

• 図書館で調べる = check at the library
• ネットで調べる = check on the internet
• 公式サイトで調べる = check on the official website

調べたら is the たら-form of 調べる.

In this sentence, it means something like:

• when I checked
• after I checked
• if you check (in other contexts)

Here the most natural meaning is when/after I checked on the official website, I quickly found out...

The たら form often connects two events:

1. first action: 公式サイトで調べたら
2. result: すぐ分かりました

So the nuance is: I checked, and as a result, I found out right away.

Because 休み here describes the state of the museum: the museum is closed.

So 博物館が休み means:

• the museum is closed
• literally, as for the museum, its status is closed/day off

The particle が marks 博物館 as the subject of the embedded clause.

This is not an object, so を would not work.

Compare:

• 博物館が休みです = The museum is closed.
• 博物館を休みます = I take a day off from the museum / I skip the museum.
  This is a completely different meaning.

休み is a noun, but in sentences like this it functions a lot like closed, off, or a holiday/day off.

In 博物館が休み, it means the museum is closed.

This is a very common Japanese pattern:

• 学校が休み = school is off / there is no school
• 店が休み = the shop is closed
• 今日は休み = today is a day off

So even though English uses an adjective like closed, Japanese often uses 休み.

かどうか means whether or not.

So:

• 博物館が休みかどうか = whether the museum was closed

It turns the whole clause into an indirect question.

Compare:

• 博物館が休みですか。 = Is the museum closed?
  (direct question)
• 博物館が休みかどうか分かりました。 = I found out whether the museum was closed.
  (embedded/indirect question)

This structure is very common:

• 行くかどうか決める = decide whether to go
• できるかどうか分からない = I don’t know whether it’s possible

In embedded clauses like this, Japanese usually does not use です before かどうか.

So you say:

• 博物館が休みかどうか

not normally:

• 博物館が休みですかどうか

That sounds unnatural.

A plain form is typically used before かどうか:

• 学生かどうか = whether someone is a student
• 静かかどうか = whether it is quiet
• 休みかどうか = whether it is closed / a day off

This is a very important pattern in Japanese grammar.

In this sentence, 分かりました is more natural because it means I understood / I found out / I learned as the result of checking.

Japanese often uses 分かる where English might use find out or learn.

• 分かりました = I found out / I came to understand
• 知りました = I came to know / I learned of

While 知る can sometimes work, 分かる is especially natural when the idea is that information became clear to you.

So すぐ分かりました means:

• I understood right away
• I quickly found out

すぐ means immediately, right away, or quickly.

It modifies 分かりました:

• すぐ分かりました = I found out right away

The sentence is saying that once the speaker checked the official website, the answer became clear very quickly.

You could move すぐ a little in speech, but this position is very natural:

• 博物館が休みかどうかすぐ分かりました。

Because the speaker is talking about a completed event:

1. they checked the official website
2. they found out the answer

So both parts are presented as past:

• 調べたら = when/after I checked
• 分かりました = I found out

Even if the museum may still be closed now, the act of checking and learning happened in the past.

Yes. Japanese often leaves out subjects when they are clear from context.

So this sentence naturally omits I:

• 公式サイトで調べたら、博物館が休みかどうかすぐ分かりました。

A natural English translation needs I, but Japanese does not.

The implied subject is probably:

• I checked the official website
• I quickly found out whether the museum was closed

This kind of omission is extremely common in Japanese.

Not as naturally in this exact form.

Because the sentence ends with 分かりました, it sounds like a report of the speaker’s own past experience:

• When I checked the official website, I quickly found out whether the museum was closed.

If you wanted a more general statement like If you check the official website, you can quickly find out..., you would more likely use forms such as:

• 公式サイトで調べると、博物館が休みかどうかすぐ分かります。
• 公式サイトで調べれば、博物館が休みかどうかすぐ分かります。

So たら can sometimes mean if, but here the past ending and context make it sound like a specific past event.