How can I maximize my chances of a professor of another university accepting to be my undergraduate research advisor?
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.