You need to express the coordinate basis vectors for the current system in terms of a linear combination of the coordinate basis vectors for spherical coordinates, and substitute into your equation. Alternatively, there is a formula for directly converting the components of your tensor from one coordinate system to another. Both methods give you the same answer.
Method 1: $$\vec{i}_x=\sin{\theta}\cos{\phi}\vec{i}_r+\cos{\theta}\cos{\phi}\vec{i}_{\theta}-sin{\phi}\vec{i}_{\phi}$$ $$\vec{i}_y=\sin{\theta}\sin{\phi}\vec{i}_r+\cos{\theta}\sin{\phi}\vec{i}_{\theta}+cos{\phi}\vec{i}_{\phi}$$ $$\vec{i}_z=\cos{\theta}\vec{i}_r-\sin{\theta}\vec{i}_{\theta}$$