yuubinbangou wo hitotu matigaete ite, yatto sonoba de kiduita.

Because 間違える is a transitive verb: it means to get something wrong / to make a mistake with something.

So:

• 郵便番号を間違える = to get the postal code wrong

The thing marked by を is the thing the mistake applies to.

A related but slightly different pattern is:

• 郵便番号が間違っている = the postal code is wrong

That version describes the postal code as being incorrect, while 郵便番号を間違える focuses more on the person making the mistake.

一つ is the general counter one thing. In this sentence, it most naturally means that one part of the postal code was wrong.

In context, English speakers will often interpret it as:

• one digit was wrong
• one number was wrong
• there was one mistake in it

Japanese sometimes uses the general counter even when English would be more specific.

If the speaker wanted to be more explicit, they might say:

• 一桁 = one digit
• 一か所 = one spot / one place

So 一つ here is natural enough, but a bit general.

〜ている often shows a continuing state or a result that remains in effect.

So:

• 間違えて = having made a mistake / and made a mistake
• 間違えていて = being in the state of having made that mistake

Here, 間違えていて suggests that the postal code had been wrong and remained wrong until the speaker noticed it.

That fits well with the second half:

• it was wrong,
• and then the speaker finally noticed it.

So the nuance is not just I made a mistake, but more like I had it wrong.

Not exactly.

• 間違えていて comes from 間違える and focuses on someone making a mistake
• 間違っていて comes from 間違っている and focuses more on the state of being wrong

Compare:

• 郵便番号を間違えていて = I had entered/gotten the postal code wrong
• 郵便番号が間違っていて = the postal code was wrong

Both can be similar in meaning in real life, but the grammar points in slightly different directions.

やっと means finally / at last / only after some delay or difficulty.

It usually carries the feeling that:

• it took longer than expected, or
• it was not easy, or
• the speaker did not notice immediately

So やっとその場で気づいた suggests something like:

• I only finally noticed right there
• I didn’t realize it at first, but eventually I did

It adds a stronger emotional nuance than a plain finally sometimes does in English.

その場 literally means that place or the spot.
With で, it means at that place / on the spot / then and there.

So その場で気づいた means the person noticed it:

• right there
• on the spot
• in that very situation

It often has an immediate, situational feel. It can refer to a physical place, but also more broadly to the scene or moment itself.

Because 気づく specifically means to notice, to become aware of, or to realize something that had escaped your attention.

That is exactly the kind of idea this sentence needs: the speaker noticed the mistake.

Compare:

• 気づく = notice, realize
• 分かる = understand, know, figure out

If you suddenly become aware of an error, 気づく is the natural choice.

It is omitted because it is obvious from context.

The speaker noticed something like:

• それに気づいた = noticed that
• 間違いに気づいた = noticed the mistake
• 郵便番号を一つ間違えていたことに気づいた = noticed that one part of the postal code was wrong

Japanese very often leaves this kind of thing unsaid when it is already clear.

Also, 気づく usually takes に, not を:

• 間違いに気づく = notice the mistake

So even though English says notice it, Japanese normally uses に for what is noticed.

Most likely, yes.

Japanese often omits the subject when it is understood from context. In a sentence like this, the natural assumption is that the speaker is talking about themselves:

• I had one part of the postal code wrong, and finally noticed it on the spot.

But grammatically, the sentence itself does not explicitly say I. Depending on context, it could also refer to someone else.

The 〜て form links the first clause to the second clause. Here it gives background or circumstances for what follows.

So the structure is roughly:

• 郵便番号を一つ間違えていて = having had one part of the postal code wrong / with one part of the postal code wrong
• やっとその場で気づいた = I finally noticed it on the spot

It is not just a simple list of two actions. The first part sets up the situation, and the second part tells the moment of realization.

Yes, slightly.

• 間違えて、... focuses more on the action: I made a mistake, and then...
• 間違えていて、... focuses more on the resulting state: it had been wrong, and then...

With 〜ていて, the sentence feels a little more natural for a mistake that remained unnoticed for some time.

So:

• 間違えて、やっと気づいた = possible, but more event-like
• 間違えていて、やっと気づいた = emphasizes that the mistake was there all along until it was noticed

Not immediately.

That is exactly what やっと tells you. The speaker did not notice it at first. They only realized it later, though still その場で.

So the nuance is:

• not at once,
• but still before leaving that place or situation,
• they eventually noticed it there.

That combination of やっと and その場で gives a sense like I eventually caught it then and there.