На этот раз я не забыл проездной дома.

На этот раз means this time.

It is a set expression:

• раз = time; occasion
• этот = this
• на этот раз = literally something like for this occasion / this time

Russian often uses fixed prepositional phrases for time expressions, and this is one of the most common ones.

Забыл is the past tense masculine singular form of забыть.

In Russian past tense, the verb agrees with the subject in gender and number:

• я забыл = I forgot if the speaker is male
• я забыла = I forgot if the speaker is female
• мы забыли = we forgot

So this sentence suggests that the speaker is male. If a woman were speaking, it would be:

На этот раз я не забыла проездной дома.

Забыть is perfective, while забывать is imperfective.

Here Russian uses perfective because the sentence is about one specific occasion:

• На этот раз already points to a single event
• не забыл means did not forget on this occasion

Compare:

• Я не забыл проездной дома. = I didn’t forget my pass at home.
• Я не забывал проездной дома. = this sounds more like I wasn’t forgetting / I didn’t use to forget the pass at home, which does not fit this situation well

So забыть is the natural choice.

Yes, originally проездной is an adjective, as in проездной билет = travel pass / transit pass.

But in everyday Russian, people very often drop билет and just say:

• проездной

So here проездной functions like a noun and means travel pass, transit pass, or monthly pass, depending on context.

This kind of adjective-used-as-a-noun is very common in Russian.

It is the direct object, so it is in the accusative case.

However, for inanimate masculine singular nouns, the accusative form is usually the same as the nominative.

So:

• nominative: проездной
• accusative: проездной

That is why the form does not change.

Russian often omits possessive words like мой when ownership is obvious from context.

So:

• Я не забыл проездной дома. = I didn’t forget my pass at home.
• Я не забыл мой проездной дома. is possible, but less natural here
• Я не забыл свой проездной дома. is also possible if you want emphasis

In English, my is usually required. In Russian, it often is not.

Here дома means at home.

It is not the plural of дом in this sentence; it is an adverb. That is why there is no preposition.

Useful comparison:

• дома = at home
• домой = homeward / to home
• в доме = in the house/building

So:

• забыл проездной дома = forgot the pass at home
• literally, forgot the pass while it was at home / left it at home

This is a very common Russian pattern.

In natural English, we would usually say left the pass at home, but Russian often expresses this with забыть что-то дома.

So the Russian idea is:

• you forgot it
• and as a result, it remained at home

That is why забыть что-то дома is a standard and natural way to say that you forgot to take something with you.

Yes, Russian word order is flexible.

The given order is natural and emphasizes на этот раз = this time.

Other possible orders include:

• Я на этот раз не забыл проездной дома.
• На этот раз я не забыл дома проездной.

These all mean roughly the same thing, but the emphasis shifts slightly:

• putting на этот раз first highlights the contrast with other times
• putting дома earlier can give a little more focus to at home

The original sentence sounds very natural.

Yes, it very often suggests that.

Because of на этот раз = this time, the sentence can imply contrast:

• maybe on other occasions the speaker did forget it
• but this time, luckily, they did not

That implication comes from context, not from grammar alone, but it is a very common nuance.

Yes. Russian forms this very straightforwardly:

• забыл = forgot
• не забыл = did not forget

Unlike English, Russian does not use a separate auxiliary like did here. The negation is just не + past tense verb.

So:

• Я забыл проездной дома. = I forgot my pass at home.
• Я не забыл проездной дома. = I didn’t forget my pass at home.