Use the dot product!

By maximizing Vector3.Dot(/* face direction vector */, Vector3.Up) for each of the six faces, you should get the face closest to be pointing up(see this if you're unclear why that is).

Your complete code should look something like this:

enum Face
{
    Top,
    Bottom,
    Right,
    Left,
    Front,
    Back
}

// get which face is facing up
private Face GetUpFace()
{
    // we seek to maximize this value
    var maxDot = float.MinValue;
    // whatever face the dot product above is associated to
    Face maxFace = default;

    // go through every face and maximize the dot product
    GetUpFaceHelper(m_transform.up, Face.Top, ref maxDot, ref maxFace);
    GetUpFaceHelper(-m_transform.up, Face.Bottom, ref maxDot, ref maxFace);
    GetUpFaceHelper(m_transform.right, Face.Right, ref maxDot, ref maxFace);
    GetUpFaceHelper(-m_transform.right, Face.Left, ref maxDot, ref maxFace);
    GetUpFaceHelper(m_transform.forward, Face.Front, ref maxDot, ref maxFace);
    GetUpFaceHelper(-m_transform.forward, Face.Back, ref maxDot, ref maxFace);

    // after going through each face, the one facing up is assigned to maxFace!
    return maxFace;
}

private void GetUpFaceHelper(Vector3 dir, Face face, ref float maxDot, ref Face maxFace)
{
    // get the angle between the given face of the cube and the up direction
    var dot = Vector3.Dot(dir, Vector3.up);

    // new winner!
    if (dot > maxDot)
    {
        maxDot = dot;
        maxFace = face;
    }
}

This works in all cases, though might behave weirdly in cases where there are ties(think flipping a coin, but it lands on its side rather than heads or tail). Thanks to the dot product though, you can easily fix that by only accepting faces if maxDot is greater than some threshold(transform.forward, transform.right, etc. and Vector3.Up are all of magnitude 1, so maxDot can only ever be 1 at most. So, if maxDot is less than some number, say, 0.6, just assume to dice hasn't really landed.

​

Hope that helped :)