There are a few problems with "How do I know I should ever see him again." The most obvious one is that it's a question, so it needs a "?" at the end, and not a period. The next is the use of the word "should". The word "should" has two popular uses; it's probably best only to use "should" in a normative sense ("You should obey the speed limit"), but people more often use it to set expectation ("You should see an "X" button in the top-left of the window"), so it's often a bit of a puzzle to know what people mean by "should". In your quote, "should" is used in yet a third way, as a synonym for "will" (a hypothetical sense). And that sense sounds pretty old-fashioned these days. So I think it's best to just say "will" for the sake of being understood. That leaves us with "How do I know I will ever see him again?", and that brings us to the final problem, which is that the sentence isn't correct English. Either of these IS correct: "How do I know that I will ever see him again?" "How do I know whether I will ever see him again?" And then there's the option "How do I know if I will ever see him again?", which is not correct, but 99.9% of native English speakers use it (either not knowing or not caring that it's incorrect), and 100% of native English speakers will understand what the intended meaning is. :)

For me, it is not a great sentence. I think it means the woman is not sure whether she will see him again or if it is a good idea. If that is the case, it would be better: How do I know if I should see him again

Hello, It means: how will the speaker ever find out, if they should ever see the man that they are talking about or not. It might be an introspective question, they seem to be asking themselves. Clear? Let me know if I helped or made things more complicated. Take care.