Just a few small notes.
Make sure the formula is mixed properly according to directions. Make sure the temperature is correct as well.
I would try the syringe again. I have no experience using a spoon. You will only be feeding a drop or two the first feeding but it will slowly increase with each feeding, you will be able to see it in the crop through the skin.
If you touch the the beak where the top and bottom beak are joined you can often get the baby to open it's beak and bob it's head, this is the feeding response that is natural for the baby and indicates that it is ready to eat.
Are you keeping the baby with the mom to keep it warm? Is she feeding it at all.
Also, if she is feeding the older baby it will sometimes work better to remove the older on to handfeed and the mom will start feeding the younger one. I am sorry I can not be of much help as it is very difficult to describe in detail something like this. Please be sure to look at the articles that I posted as they give many detail that you need.
Keeping the baby warm and at the right temp is important and you just have to keep trying. It sometimes takes the baby time to learn to eat well. It just depends on the baby. You can also try putting some formula on the left side of the baby's beak just above the opening and see if it will run into the babies mouth and if he will eat it. I am not sure this is the recommended way but it seems to work for me with very young babies. It is preferable to do this with a baby who is exhibiting a feeding response.