Will's answer is excellent and very thorough.
However, if you'd like to take it a step further, the fact that carbon monoxide only acts as a reducing agent does not exclude its participation in a species' metabolism. The trouble is that the organism would have to have a wildly different metabolism, meaning that it would have to be evolved from a common ancestor much further down the evolutionary tree.
Most of the organisms we've found with strange metabolisms (called extremophiles) are in the archaea family. It would be conceivable (although of course unlikely) that a multicellular organism had evolved within the archaea rather than eukaryote domain.
One of the most consequential moments in the evolution of life on our planet were the mutual endosymbioses between chloroplasts and cyanobacteria, and between mitochondria and proteobacteria. In the world you're building, it would be possible that there was an endosymbiosis involving an archaea, such that the multicellular organism's metabolism evolved around the symbiosis of an archaea.
The idea of the mitochondria is that it generates ATP through pumps that exploit a chemiosmotic potential across its membrane. This potential can be established without having to start from the same reagents.
To borrow from Will's answer, you could for example have a metabolic pathway that starts from carbon monoxide using CODH
$CO+H_2O \rightarrow CO_2+2H^+$
You could thus have a buildup of $H^+$, establishing a gradient just like in a mitochondrion. The $CO_2$ produced in this reaction could also be used to produce methane, which would establish yet another gradient.
$CO_2+4H_2 \rightarrow CH_4+2H_2O$
In summary, the existence of the organism you are describing is totally plausible, but you would have to tie it in with an alternative evolutionary branch. Do keep in mind that if you try to approach the problem in extreme detail, you may find yourself digging a deep hole. We don't yet fully understand real metabolism, so to try and invent a complete metabolic system would be a mammoth of a task, given the incredible complexity and interconnectedness of metabolic pathways.
Good luck with your little carbon monoxide breather!