It is not possible to unwrap an arbitrary fisheye image to the cubemap. However, if you were talking about the spherical environment map (SEM), but not fisheye image, then it can be done with some limitations.
Some time ago, I've managed to write a software that could convert a SEM map to cubemap and vice versa. So I can prove that it is possible.
You should be aware, that you would get large distortions in the region close to the edge of the SEM, that is mapped to the center of back face of cubemap.
Read this article, you will find there all needed information to implement what you want.
There is a function that maps the direction vector to uv coordinates of the SEM on page #9. On page #21 you will find similar function, but for the cubemap.
So, your strategy would be to find for every texel of cubemap a direction vector, then for this direction vector calculate SEM coordinates and sample a texel from the SEM map.
P.S. From the link, that you've added in your comments, I've notice that you were talking about a half sphere environment maps. All the logic would be the same, except the mapping from 3d direction vector to SEM coordinates. It's not hard to change the equation from the page #9 in order to work with half sphere. You should be aware, that for a half-sphere SEM you will get a half cubemap. I mean, without the back face and with cropped side faces.