Should I talk about a unpublished paper in my SOP?
Talking about an unpublished article that has inspired you is certainly fair game for a SOP essay in a graduate application.

Certainly the inspiration, and the differences between your approach and different, existing methods would be highly interesting to me.

I love to see undergrads who take initiative in research - they tend to make good Ph.D. students. Taking ownership and initiative in your project is a key step of maturity in grad school. So if you have already done so, that's a great sign.

And yes, if it's only in Arxiv and you haven't mentioned it to your letter-writers, I'm not likely to read the paper if it's not otherwise mentioned in your SOP.