In recent years, gas has been observed in an increasing number of debris discs, though its nature remains to be determined. Here, we analyse CO molecular excitation in optically thin debris discs, and search ALMA Cycle-0 data for CO J=3-2 emission in the Fomalhaut ring. No significant line emission is observed; we set a 3-$sigma$ upper limit on the integrated line flux of 0.16 Jy km s$^{-1}$. We show a significant dependency of the CO excitation on the density of collisional partners $n$, on the gas kinetic temperature $T_k$ and on the ambient radiation field $J$, suggesting that assumptions widely used for protoplanetary discs (e.g. LTE) do not necessarily apply to their low density debris counterparts. When applied to the Fomalhaut ring, we consider a primordial origin scenario where H$_2$ dominates collisional excitation of CO, and a secondary origin scenario dominated by e$^-$ and H$_2$O. In either scenario, we obtain a strict upper limit on the CO mass of 4.9 $times$ 10$^{-4}$ M$_{oplus}$. This arises in the non-LTE regime, where the excitation of the molecule is determined solely by the well-known radiation field. In the secondary scenario, assuming any CO present to be in steady state allows us to set an upper limit of $sim$55% on the CO/H$_2$O ice ratio in the parent planetesimals. This could drop to $sim$3% if LTE applies, covering the range observed in Solar System comets (0.4-30%). Finally, in light of our analysis, we present prospects for CO detection and characterisation in debris discs with ALMA.
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