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Good questions!  Yes I think that would absolutely work, since the important part is having a shared experience. To be clear, the only reason i was saying same sport was so that you would have a common interest/ experience. My experience with networking is that the more you have in common with someone, the more likely they are to respond, help out, etc. That being said, it also can work if the only thing you share is being an alum. Bonus points if you have friends who graduated a couple years ago who are now at MBB. Given what you are saying about your school, however, it sounds like a relatively tight-knit community of MBB alums, so i would explore any connection you can find.

I would just play around with linkedin, you can even do google searches with "site:linkedin.com". I would also recommend looking at some of the other wso content on networking to get an idea for conversation points, but largely speaking, you likely want to ask them about the job, their pathway, what cases they've worked on, etc. I'd also recommend that if you find someone who is focused on something you don't have knowledge of, do a bit of research on the firm's practice in that industry, or use part of the conversation as an opportunity to learn more about the industry.