This order is correct. While the common order of French sentences is Subject – Verb – Complement, here is one of the cases where the opposite order is possible, especially in a literary context.

As Christophe rightly points, there is a difference between au premier coup and du premier coup.

Here the sentence has a more than likely implicit part:

Au premier coup [de hache] ne tombe pas l’arbre.

i.e. “with a single [axe] hit, the tree doesn’t fall down” or “on the first [axe] hit, the tree doesn’t fall down.”

The verb is intransitive and the first clause is an indirect object or circumstantial.

See this page that gives some examples like:

Autour de l’arbre s’était enroulé un chèvrefeuille

vs

Un chèvrefeuille s’était enroulé autour de l’arbre

and

Une heure plus tard se produisit un évènement imprévu

vs

Un événement imprévu se produisit une heure plus tard

The usual form is subject + verb + complement:

L’arbre ne tombe pas au premier coup.

You can insist on the timing by starting with the complement:

Au premier coup l’arbre ne tombe pas.

When you start the sentence with something else than the subject, you may invert the subject and the verb. It is a stylistic figure:

Au premier coup ne tombe pas l’arbre

CAUTION: there’s a difference between “au premier coup” and “du premier coup”. The former takes “coup” in its literal meaning (shot, stroke, blow, glance) whereas the latter is figurative (i.e. the first try).