As Pedro said, geothermal/seismic activity would be a good way of the planet generating its own heat. I want to address the other questions:
Would radiation from the sun spill over the sides?
I'm less able to answer this question, but I think it depends on the ringworld itself. For example, the Earth's magnetosphere deflects a lot of solar radiation around the Earth, so assuming a similar magnetosphere (magnetoroid?) surrounds the ringworld, it's likely that some radiation will be deflected around the ring. However, it's not likely to be enough to heat the planet.
Could the metallic ringworld radiate an appreciable amount of energy from its dark side into space?
If we assume the ringworld is an ideal black body (emissivity of 1), and has a width equal to the diameter of Earth, and transmits the amount power received by the Earth from our own sun from a square on its surface 7914m x 7914m (diameter of the Earth squared):
1.74x10^18 W = (1)*sigma*62.69x10^6 m^2 *T^4
The surface of the Ringworld would have to be around 26450 K. About 4.5x the sun's surface temperature. This is a huge approximation (trying to roughly account for how much of the radiation our planet might receive from the ringworld), but it doesn't seem plausible. If the ringworld in question were considerably larger in size (mostly width) and thus able to absorb and radiate more heat via greater surface area, it might approach the "possible", but I don't think it would be enough on its own to heat a planet.