When you are lonely, it feels like the world is against you, that the world tries to prevent you from being happy and doing what you want to do. This is the initial point of the text and it would have been better to have put this phrase at the beginning of the text, in my opinion. "That way" means "in that way". It means "using this method". The "way" previously mentioned in this text is being with someone who understands you. "(in) that way" develops an argument: you define a strategy, plan, or method. Then you introduce a description of the effects of using that method with "(in) that way".

Note that "you" does not have to refer to the person who you're talking to. It can also refer to "just any person" and in this case, it really sounds to me like he's talking about a general idea. So, not "Wouldn't YOU like to not be alone?", but rather "Wouldn't it be better to not be alone?" (I just did the same thing. I said, "to the person who you're talking to" - that "you" does not mean you, midnight - at least not only. I could have written "someone" or "the speaker" as well.) And because of the topic Patrick is talking about, I even think he's talking about himself. Since "Patrick" has just explained that he wants to have a partner for his problems and his life, I'd suggest to understand the "you" as a plural form. The sentence is famous and normally it would be singular. It's not quite the same as saying "The world is against you". This would only say that your situation is bad, no one supports you and probably you have bad luck. "You against the world" also includes the idea that you fight back. It's like in sports: This game is "London against Berlin", or "You against the whole team", or in this case, you against the world. It's a romantic term, too. Imagine that you can cope with an opponent like this. So, Patrick wants to play this game in team with someone else. You against the world = He and his imaginary friend/partner/love against the world. If it's plural, this is like "We against the world", and this is even more romantic. This one also implies unbreakable loyalty through all hardships. And you concentrate on this one relation, everyone else is far less important than the partner.

You could imply that the world is against him or her, but it's not clearly stated. The text is not of a high quality and so analysing it in a lot of depth is not a useful exercise, in my view.