What is the purpose of the inverted gravity?

The gravity on the ship is not inverted, it is redirected to allow the humans to walk through the tube that is the ship.

The most technically challenging sequence has Banks and Donnelly entering the alien ship for the first time, through a tunnel in which gravity disappears, allowing them to walk vertically up the shaft.

While the gravity shifts in such a way that it conveniently allows the humans to walk up the ship, avoiding any need for climbing, stairs or elevator like structures, it's not entirely clear whether this is purposeful design by the Heptapods just to assist the Humans.

To answer your question about it relating, or symbolizing the non-linear time plot of the movie, I'm not sure that anything really supports that.

Humans, and presumably most creatures that evolve on a planetary structure, require gravity to assist with their health long terms. Humans without gravity suffer muscle and bone density loss, even with exercise. If artificial gravity is possible, and engineeringly feasible, the designers of an interplanetary craft would use it for the health of the passengers.

The fact that the internal gravity doesn't align with the Earth's field doesn't seem particularly surprising.

It may be advantageous in other ways to be in that orientation while traveling in these craft. It does seem a little clumsy as a mechanism to enter and exit the craft on a planet since you would suddenly find yourself dropping if walking out, but who knows what docking structures its designed to normally work with.

It also does make a fun plot point from an out-of-universe perspective.

The ability to experience non-linear time is definitely not related to the ship's artificial gravity. For one thing:

Louise still experiences non-linear time even when she's not on board the ship, in the artificial gravity it creates. Indeed, she still experiences it for years after the heptapods have left.

The fact that a space ship capable of travelling interstellar distances has artificial gravity is not too surprising. That they chose to orient their ships in a way that did not align with Earth's gravitational field may seem odd, but if you think about it, it's just another puzzle to solve.

The heptapods need humans to solve a puzzle. Their crisis for one, but of more immediate concern, they need humans to solve the puzzle of how to communicate with them. They can't just speak English at us, even though there is some evidence they know the language, because:

they need us to learn their language.

And necessity is the mother of invention, so to speak. If we knew they could understand English (or any other language from Earth), we might stop trying.

They don't want to deal with every random yokel who might wander in if they make it too easy to gain access, so they make that hard to ensure that the people who do come will be ready to solve puzzles and learn.