That could work if your villain is on a budget, but a time frame of a few centuries is MUCH too optimistic. It would expect diverting even a small asteroid to take several million years, let alone one of the really dangerous ones.

Making the sail (much) bigger would help, but then you have additional problems because people would see it coming and could take countermeasures.

If your math is good enough, could you just "paint" the surface of a suitable object enough to steer it via differences of light pressure?

Not really. The differences in pressure would create spin, not push the craft in a different direction. You would need something else to push against, to "tack" against the solar wind, like how a sailboat's keel pushed against the water. Magnetic fields are an option, but those get less and less as you move away from the sun.

There are some complicated options involving mirrors that could move a craft in other directions, but they are radically difficult to manage. You send five mirrors with the craft as a formation, one on each side and one ahead. Then you can bounce the lasers off these mirrors to steer the ship. At the destination, the ahead mirror is used to slow the craft. All these mirrors would essentially be separate craft that would need to be themselves accelerated and managed.

If you use a material on the back of the asteroid that had a higher albedo at more oblique angles, you could have a form of feedback.

Interestingly, this is one of the ways we could deflect an asteroid on a collision course with the Earth, if we detect it early enough.

Yes, some deep space missions need to take solar radiation pressure into account due to effects like this.

In general, the answer to "If I ignore how physics works, can I XXXX" is "yes", because what you've actually asked is "can I do XXXX with magic".