How do I approach a professor from another university?
I'm playing professor psychology here; may serve as another approach for you to think about.

If it's an experienced professor and the school is large, your cold calling e-mail may not stand out (unless you have some stellar skill sets or experiences.) If it's a relatively new professor, hiring someone outside his/her school may be a deterrence.

My suggestion is this: talk to your school to set up a cross-registration arrangement and take a course at the state university. Preferably, take a course that is taught by the professor with whom you would like to work.

This gives you a few edges: you'd get to see if commuting to two campuses work for you, you'd get to check out all the resources and feel the campus culture there, and you will get to know the professor better, and be able to make a personal contact with him/her. Get a good grade, and use the semester to build a relationship with the faculty there.

Once the professor got to know you and understand your aspiration and capability, it'd be a lot easier for the professor to engage and make a decision.