How can I maximize my chances of a professor of another university accepting to be my undergraduate research advisor? Next year I'm going to college and would like to do undergraduate research in the second semester under the supervision of a certain professor, who I'll call Prof. M.

There are three points against me and three in my favor. They are the following


Prof. M. works at another university, which is about 100 miles away from the one I'm going to. If he does accept to advise me, my plan is to travel to his university every week or every other week.
I am worse than terrible when it comes to social aspects: I talk too fast, lack common sense, and I feel horribly nervous when talking to other people (specially people I don't know), often leading to awkward behavior for not knowing how to act.
Prof. M. is a (local) bigshot. (I'm assuming he might be more selective because of this)
I have self-studied math and physics more or less to have approximately the technical understanding a second year graduate student would have. Of course, while I know about e.g. locally ringed spaces or bosonic string theory (or insert buzzword here), I lack most of the skills and experience a real graduate student would have, such as being able to critically read research papers or having research experience. Also, this is a self-assessment, thus possibly a faulty one;
The professor is welcoming and has a nice personality (I got this information from a friend that has a friend who was a phd student in the same university as Prof. M. and who talked to him personally);
He has supervised* undergraduate students in the past (about five years ago);


So,


should I talk to Prof. M. personally or by email?
should I mention a paper he wrote that I find particularly
interesting and ask if he would accept advising me to do similar
research?


In short, what is the best path to be taken here?



*Though all undergraduate students are from his university. See Professor Eldredge's comment below.
For reasons completely unrelated to your own skills and background, I think your chances  here are slim. The professor has little external incentive to take you on and doing so will complicate his/her already too busy academic life. There might even be rules forbidding it, though the chance of that is small. 

But, you ask how you can maximize your chances, so I'll give some hints. 

This is the sort of thing you want to do in person, not by mail/email. You certainly won't get a positive answer via an email exchange. You might get an invitation to visit, but that puts you face-to-face in any case. 

This is the sort of thing you want to do in conjunction with an advocate/intermediary. If you were already established in "your" college, one of your professors would be good for doing that. Especially since the prof once studied there. But if you have a current teacher who knows this person (or is known) they might be willing to intervene on your behalf, perhaps even accompanying you on a visit to the other university. 

I'm not sure that waiting until you are established at university is a good thing or not. You won't have a lot of time to become recognized before you need to make the contact. But it would give you the potential for finding an intermediary.

You will need to convince the prof that you have unique skills and that it would be a missed opportunity not to work with you. That is a pretty steep climb. It would help if some of your research interests align. So you need to demonstrate that you know something of the research of the prof and how you might fit in. 

But I also have a small wonder about why you chose the university you did, rather than the other one. You would maximize your chance by going there instead. 

Don't worry about being/seeming awkward. Lots of people in academia are likewise.