Learn Turkish: Yeni bir temizleyici aldın mı?

The verb root is temizle- (to clean), which itself comes from the adjective temiz (clean). Since the root ends in a vowel, we add the buffer consonant -y- before attaching the agentive suffix -ici. This gives us temizleyici, meaning someone or something that cleans.

Does temizleyici mean a cleaning product or a person who cleans?

It can be either! The suffix -(y)Icı creates a noun for whatever performs the action. Depending on the context, temizleyici can refer to a person (a cleaner or janitor) or a product (a household detergent or a facial cleanser). In this sentence, it most likely refers to a cleaning product.

Why is the word order yeni bir temizleyici instead of bir yeni temizleyici?

In Turkish, when you use an adjective with the indefinite article bir, the adjective almost always comes first. The standard order is Adjective + bir + Noun. Saying bir yeni temizleyici directly translates the English word order but sounds unnatural to native speakers.

Why doesn't temizleyici take the accusative case marker here, since it is the direct object of the verb aldın?

Because it is an indefinite object. The presence of bir makes it 'a new cleaner' rather than 'the new cleaner'. In Turkish, only specific, definite direct objects take the accusative case marker, so the noun remains in its bare dictionary form here.