Changing from AI to CDR is a bad idea. That should not be done for "delivery" to the printer. That only should be done if the other person is going to work on the file for future modifications, (including re-processing a bit the artwork to be simmilar again)
Normally, for simple vector artwork (Logos, flat images), the delivery format should be pdf. To be more specific X1 or X3a subformat.
But this kind of effects should be flattened on a raster image. In fact the subformats X1 and X3a do that. Do not allow transparencies, or blending modes, etc. This kind of effects are flattened.
You could export directly to a pdf, but I like to be more specific and prepare a copy of the document, and rasterize specific layers to control the conversions.
If the print is going to be digital, you can convert to an RGB file. If it is going to be offset print, It should be CMYK, but if your artwork was RGB expect some changes. It would be better if you prepare thoose changes in the Vector Artwork itself.
If the other person is going to be using this image in several projects in Corel, you can prepare a higher resolution raster image, so you he/she will not be limited by the resolution if he needs to scale it up.
As a last resource, the exported image should be tweeked inside Corel to restore the original appearance, using the internal gradients and blending modes.
One aditional thing you could try is saving the AI file using different settings, for example using a previous version of ilustrator (1 or two versions) and turning on and off the pdf compatibility.
As one last note, you are also having issues with the color modes. I have the feeling your AI image is using a diferent color profile.