Learn Russian: Доплата будет максимум долларов десять. - The surcharge will be a maximum of about ten dollars.

This is how Russian shows approximation! By putting the noun (долларов) before the number (десять), it changes the meaning from "exactly ten dollars" to "about ten dollars."

Does flipping the word order change the case of the noun?

No, the number still controls the case of the noun just like normal. Since the number десять requires the genitive plural, доллар becomes долларов even when it is moved to the front.

Why use the inverted word order when we could just use a word like около (about)?

Flipping the word order is very natural and common in everyday spoken Russian. It is also grammatically easier: if you use около, it forces the number into the genitive case (около десяти долларов). By just swapping the words, you can keep десять in its basic nominative/accusative form.

How does максимум fit into the sentence? Does it need to match the noun?

In this sentence, максимум acts as an invariable adverb meaning "at most." Because it is functioning as an adverb and not an adjective, it does not change its ending to match the feminine noun доплата.