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So going through all the shown Servants, there are a lot that we have info on, Jekyll, Scathach, Darius, and Arjuna. Of those, we have the most information about Jekyll because he's in Fragments. Scathach is mentioned in HA and Darius is mentioned in Zero. Arjuna is actually mused upon quite a bit by Karna since he is the main driving factor for Karna, at least in Apo.
Either way, here I'm going to try summarize the relationship between Karna and Arjuna in Karna's eyes, since that's where the perspective is from. I have no experience in Indian Mythology so yeah. This will be spoilers for Vol 4 and Vol 5 of Apocrypha.

So as we all know Karna was born and given his sun powers by his dad and his mother got his armor for him though her pleading. However, these are things that Karna never wished for. Even so, he is grateful of such an honor. Therefore, after his mother abandoned him he moral compass has always be "I won't disgrace my father." And that's why he's such a great guy.
However, the only person who's ever disturbed Karna was Arjuna. Arjuna was a someone loved by everyone, and called Kiriti, Vijaya, Dhananjaya, etc. And in a way, you could say that Arjuna was the person who took everything that Karna had, his armor, his spear, what-not. So basically everything that Karna got from his parents Arjuna took it away.
So then Karna thinks that he should be jealous of such a person, but for some reason he can't put his finger on, he's not jealous, but rather during the last moments of his life, when Arjuna shot him, Karna was disturbed.

Karna doesn't understand this, but he thinks that Siegfried is something who is close to Arjuna. Therefore, by fighting Siegfried he can understand what he felt during that final battle. However, Siegfried is basically dead. So Karna's okay with fighting Sieg to find out the answer to his question.

So then when he's fighting Sieg he figures it out. Well first, he figures out his own wish. Karna was someone who was bullied and what not; however, the only time he felt he was equal to everyone else, or even better than everyone else was on the battlefield fighting. However, even if that was his time to shine, he had all these invisible restraints but on him. Like "don't fight anyone other than Arjuna," etc. However, because he is summoned as a Servant he is no longer bound by fate he can fight without any restraints anymore. And against someone like Sieg who won't die easily because Armor of Fanfir he can fight his fullest. At this point he realizes that his wish was to beat Arjuna, to win. So for once in his life, actually come out on top when going all-out no bars. Which is why he goes VS against Sieg.

So then after he loses. He realizes that what he felt in his last moments, when Arjuna shot him, was actually happiness. He's happy that Arjuna broke the Warrior's Code and shot him. That for once Arjuna showed a crack in his perfect persona and did something that Arjuna actually wanted to do. Which is why Arjuna was similar to Siegfried. They both did whatever was asked to them; however, inside they had their own wishes. Siegfried's of course being a Hero of Justice. No idea what Arjuna's was but maybe he's secretly a psychopath?

Either way, what we can gain from this is that Arjuna in Grand Order is a guy who acts like he's perfect; however has an angry side. We can see this with his third voice clip when he's like "you saw me!" like you saw his true nature, the one he keeps behind his persona. However, I have no idea why his wish would be eternal solitude.

Basically, that's my prediction on how Grand Order Arjuna is going to be like based on what we got. I really hope I roll him, so I can see if this is true or not.