"The girl is flying in July."
Translation:A lány júliusban repül.
Yes, it is. It emphasizes the girl, though ("The girl is flying in July." or "It is the girl who's flying in July.".)
And A lány júliusban repül presumably emphasises "The girl is flying in July" or "It's in July that the girl is flying", and Repül a lány júliusban presumably emphasises "The girl is flying in July (not taking the train)".
Is there no neutral version that does not emphasise anything in particular?
There isn't necessarily an emphasized part of a sentence. When the place of the emphasis is pointed out by me & other users, the alternative word order is unusual, which would most of the time imply that the focus isn't empty. One way of determining wether there is a focus is to check wether the word prefix is separated from the verb. They are seprated if there is something in the focus (unless it's the verb prefix itself.)
Júliusban elrepül a lány.
Júliusban repül el a lány.
In the second sentence I'm really stressing that it's July, that's the whole point of why I'm talking/writing (The rest of the sentence isn't just fluff, though, you can say for example: "Júliusban repül el a lány, a fiú augusztusban.")
When there's no verb prefix, it gets ambiguous in writing (not in speech, since the focus (if there is one) is stressed.) There's no ultimate rule to get a usual word order when the focus is empty, it comes with practice. In this particular case, the most natural thing to say is "A lány júliusban repül."
Let me take this opportunity to sneak another Hungarian suffix in here: the translative.
A lány óvónővé válik augusztusban. - The girl becomes a kindergarten teacher in August.
The suffix is -vá/-vé (the v gets assimilated after consonants) and is always used when something becomes something else. Jézus átváltoztatta a vizet borrá. - Jesus turned (the) water into wine.