Save the game before entering the large room beyond the crawlspace. Once you trigger the cutscene in which the mechanical horseman appears, you will have one chance to try this strategy. If it doesn't work, you'll need to reload and try again.
Step forward carefully until Lara is standing just inside the doorway to the big room. The seam between the tiles is marked with a broken line in the screenshot above. Stepping over this line triggers the cutscene, so don't do that yet.
Equip the grenade launcher (with regular or super ammo; flash grenades won't work) and pivot in place until Lara is facing the alcove in the northwest corner, just to the right of the bronze horse.
Fire a grenade and immediately step forward to trigger the cutscene.
If you've done it correctly, the grenade will land on the floor in front of the mechanical man, just as he emerges from the alcove...
...and you'll see it explode during the cutscene. This is the only time the mechanical man is vulnerable to explosives.
Now, instead of mounting the bronze horse and charging at Lara, as he would normally do, he just paces around the horse.
Before he turns his attention to Lara, switch to the shotgun (or pistols/Uzis), run out into the open room, position Lara behind the horse, and turn to face the horseman.
Now you have plenty of room to back up as you fight him just like you did the mechanical men you encountered earlier.
When you defeat him, both horse and man disappear, leaving behind the HORSEMAN'S GEM. Pick it up.
NOTE: As far as I know, this method works on all systems, though it does require fairly precise timing to fire the grenade just before triggering the cutscene. If you can't make it work, if you didn't happen to save before entering the room, or if you're out of grenades, see the main walkthrough for alternatives.