Determine the quantitative composition of a solid sample containing Co, Mn, Al, C, and O
It very much depends on element concentrations and the accuracy required for the total assay, but, I would do the metals by ICP-AES or AA and C by a classical combustion technique. ICP-MS would be unnecessarily expensive and since you are looking at compositional levels rather than low trace levels, it just wouldn't buy you anything.  

If you know oxygen is the only other element present at compositional (i.e. percent) levels then don't bother with the finickier oxygen analysis and just determine it by difference. If carbon is present at at least a few percent then you only need a few mg, leaving enough to do multiple digestions for the metals analysis if desired.  

If you want to do direct oxygen analysis, it can be done by classical pyrolysis techniques, though this presumably high-metals matrix would require a special technique (Mertz) and this analysis could have relative uncertainties of greater than 10%.