Arnold simply translates the color of the particles, so you need to setup your simulation in Cinema 4D to inherit the color from the mesh. This means getting the color based on an Arnold shader is not possible, since that shader is evaluated in render-time and Cinema 4D does not know anything about it.
If you need something 'simple' and just get the color based on a texture, it should be doable. Attached is a simple setup with a Plane, using a Shader Field to transfer a texture (a Gradient shader in this case) to a vertex map, that is mapped to the particles.
Otherwise you can achieve what the Cinema 4D particle system allows.
// Peter Horvath
// C4DtoA developer
