What is the significance of the artwork?

The Painting is Christina's World by the American painter Andrew Wyeth.

I think it is supposed to be one of the works of art saved by the survivors, Why Julia had it at the end (I think Jack put it in her sleep capsule, but can't recall 100%) is that Julia said it reminded her of home.

Now first of all, the painting itself is Christina's World from Andrew Wyeth.

The interesting part here is, though, that it doesn't just show a woman lying on the grass but this woman actually suffers from muscular deterioration and tries to make her way across the grass using her arms. In his analysis of the movie (unfortunately not available in English) Wolfgang Schmitt elaborates, that while her body is disabled, she is still free in her mind and can thus shape her own world. He sets this in relation to the questions about humanity that the movie supposedly asks and the conflict between the analog and digital world and sees it as a way to overcome this conflict and for the human to ultimately find his place in the world, the home he longs for (translation by me):

The base question that the movie asks is "what is the human?". The movie then gives many, and some very contradictory, answers to that.

One answer is "the human, that's someone who lives in the nature, who lives in unity with nature, who's at home in an analog world." Again and agian we see how Tom Cruise visits a place of idyll, a place of longing. [...] But at the same time and while the camera is so in love with this place, so in love it also is with the futuristic architecture of the base from which Tom Cruise is controlled from and to which he always returns. [...] The movie is insecure of what it actually wants and that speaks for our time. [...] We all again love the real, the autentic, but only as long as we don't have to abstain from the artificial...

Of course this escapism always stays ambivalent. It is a double-edged blade, we can never completely return to nature and we don't want to completely dissolve into the didigtal either. But the movie also gives another possibility. [...] The scavengers aren't really scavengers but they're actually the last humans on planet earth. And they live in a kind of museum...

And there's one painting, that's shown again and again, Christina's World from Andrew Wyeth. [...] On this painting we see a disabled girl lying in a field and trying to make her way with her arms, for she wants to a house that's seen in the background. We see here a human imprisoned in his body, disabled actually. The human is seen here as a being of deficiencies, as Gehlen put it, and the human is always a being of deficiencies, even when he's not disabled. But in this painting there's even more. It's a symbol for the human per se, because the human wants to be home somewhere in the world. And he wants to escape his body while at the same time being bound to it. But he also is, and that's interesting, a being of spirit, who can salvage himself into freedom by the power of thoughts even when he's disabled. That's the special position the human has. The animal has to adapt to nature, the human can change the nature. But because of his deficiencies he's also bound to a human tradition. Bound, but not imprisoned, he can move inside this tradition, this culture, and he can develop this culture further...

The painting is particularly important in interpreting the movie. As imprisoned this woman is in her body, as free and cosmopolitan she also is. She isn't imprisoned in the world, she has world because her mind is free. This goes as far in the movie in that also the martyr's death becomes a topic. There the human also is free in the most extreme position because he can take his own life himself. The film thus actually shows here what is inside the human.