He might rely on his powers. Percy is a son of Poseidon, so his control over water could be crucial. He could use water sources in Tartarus, if any, to heal himself, create barriers, or even find a way to escape. For example, he could manipulate the Styx water to his advantage.
Alliances. He could form alliances with other beings in Tartarus. There might be some creatures or even fallen heroes who are also against the forces of Tartarus. Together, they could share knowledge and resources to help Percy survive. Maybe he teams up with a powerful Titan who has a grudge against the current rulers of Tartarus.
He might rely on his skills and the powers he already has. His control over water could help him find sources of water in Tartarus for survival. Also, his combat skills would be crucial in fending off the various monsters there.
They might survive by relying on their combat skills. Percy, with his sword - fighting abilities and Zoe's archery skills, can fend off the various monsters in Tartarus. Also, they could use the environment to their advantage. For example, hiding in the dark corners or using the natural formations as shelters.
First of all, Percy's resourcefulness would come into play. He would probably try to make allies among the less - hostile creatures in Tartarus. He has a way of befriending those who are often overlooked. Also, he would use his knowledge of Greek mythology. Since he knows about many of the monsters and their weaknesses from his studies, he can use this knowledge to survive. His connection to his friends, even if they are far away, might also give him the strength and motivation to keep going and find a way to escape or at least endure.
Percy could initially be heartbroken. He's a very loyal person, and to be betrayed by those he considered family would be devastating. In Tartarus, he would have to rely on his instincts and his powers. He might also experience a lot of self - doubt, wondering if he really was as bad as they made him out to be. As time passes, he would start to plan his revenge against those who wronged him while also looking for an escape route.
Surviving in Tartarus would require them to adapt quickly. The place is full of unpredictable and extreme conditions. They need to find shelter from the various hazards, like the volcanic eruptions and the attacks of powerful monsters. Their friendship and mutual trust would also be a great strength. They can support each other emotionally, which is crucial in such a terrifying and desolate place. For example, when one of them is feeling down or hopeless, the other can cheer them up and keep their spirits high, so they can keep on fighting for survival.
Well, one reason could be that he's wrongly accused of a major crime against the gods. Maybe some false evidence is planted, making the gods believe he's a threat, so they banish him. Another reason might be that he's trying to protect someone or something, and his actions are misinterpreted as going against the gods' will. For example, if he's trying to save a powerful but 'forbidden' creature, the gods might see this as a violation and send him to Tartarus.
Perhaps a prophecy misinterpretation. If a prophecy was misread and it seemed like Percy was the cause of great evil to come, the gods might take drastic action. For instance, if the prophecy spoke of a 'child of the sea' causing destruction, and they wrongly assumed it was Percy without fully understanding the context.
He uses his wits. Percy has always been clever, and in Tartarus, he'd use that to his advantage. He'd find ways to outsmart the monsters, like using the environment against them.
He could rely on his natural demigod abilities. His control over water, for instance, might help him find sources of water in Tartarus that are safe to drink or use for protection. Also, his combat skills, which he has honed over many battles, would be crucial in fending off the various monsters in Tartarus.
In the fanfiction, the environment of Tartarus itself could be considered an antagonist. The extreme heat, the toxic gases, and the ever - present danger of falling into bottomless pits. And on top of that, there could be some sort of demon - like creatures that are specifically created or summoned to hunt down Percy and Zoe and prevent them from achieving their goals.