Your interpretation that it sounds as if the phrase is talking about "a general schoolboy" is actually fairly accurate. "Something of the schoolboy" is a specific form of the general phrase that can be used with any noun, thus: "something of the X." For "X" you can subsitute any noun. "There is something of the devil in that boy" would be just one example.
This is a very well known phrase that means: Possessing a measure of the defining characteristics of some other entity. If I say there is something of "the jungle" in that city, it means there are characteristics of that city which would remind you of the jungle.
Here's an example to make it a little more clear. Let's say you are talking to a person who is a secretary, and not known to do anything out of the ordinary. You say to that person, "What are you going to do this weekend?" And that person says, "I think I'm going to go skydiving this weekend." You smile and say, "There's something of the daredevil in you after all." It means "you are slightly daring after all," or "even though you are a usually boring secretary, it turns out you do have a bit of daring in you."
Something (a portion, part, or characteristic) of the daredevil (the type of person who is daring) is what this phrase indicates. "You have something of the daredevil in you" equates to "you are slightly daring." "You have something of the schoolboy in you" equates to "you are slightly schoolboyish."
So there you have it in a nutshell. "Something of A schoolboy" is not the way it is usually said. The only way I can see "something of a schoolboy" being used, in fact, would be if a person were acting completely like a schoolboy, when that person was not actually a schoolboy, and then you might comment, "He's something of a schoolboy, isn't he?"
I'm tacking on an edit now because a light bulb has just gone off above my head. A syntax variation has recommended itself to me as being influential in whether "a" or "the" works better. I find myself more comfortable with "the" when the phrase continues, thus: "something of the beast in the man." But when the thought is completed without the need of the continuation of the phrase, "a" works well, thus: "the man is something of a beast." Now, admittedly, I think there is a subtle distinction between the meanings of these two locutions; the first would be "he's a man but he does have some beastly characteristics," and the second would be "he's really quite beastly (more so than I think he should be)." But at least I think these allow for the possible use of both "a" and "the."