I played through the game on hard, and thought i would have trouble with this boss. But, it actually was very easy, this is how I did it:
Make sure you have atleast 5 bars of health before boarding the Athena. When you first enter the room, (after switching on the belt with a crate on) move left behind the crate on the floor, and wait for the first alpha drone to move towards the convayor belt (Spinner will tell it to check out the crate). Then keep moving round the room to the other side where a majority of the shadows and cover is. Kill the alpha drone, then imediantly from the area you came from, anouther will spawn. Move back into the shadows whre you kill the first drone then repeat for the second time.
After this phase of the fight, i was left with 3 bars of health. I crouched down behind one of the drones closest to the large door on the side of Athena and faced towards it. You will have about 30 seconds to position yourself and start throwing grenades from the drone at the center of the door.
As soon as the door opens, keep throwing grenades and after the second one explodes on Spinner, it will kill him.
If you time this right, you should kill him without loosing any more health, otherwise .. keep throwing untill one of you die!!
Good luck!!