Instead of spread plate method in different media for determining bacteria and fungus colonies . can we determine overall cell viability (microbial cells as other raw material cells are dead or decomposed) by MTT assay?
The MTT (as other tertazolium salts) is an enzymatic assay focusing on reductases activity (found in metabolically active cells). Be aware that using the MTT will give you an estimate of the total reductase activity in your compost. Up to a certain point (again be very careful) this could be consideed as a proxy (and only as a proxy) for the microbial load and eventually viability.
This said your compost probably contains highly active, less active and probably inactive but still viable microorganisms. Therefore your microbial load estimate will be biased in one way or another.
In conclusion if your are looking at microbial load (the amount of living cells per amount of compost), you should use plating (alternatively PLFA or another quantitive technique mighte be useful)
If you are interested in viability, then you should consider live/dead staining.
In the end, it is your choice to choose wether MTT provides interesting informations for you. But please be careful in the interpretation of results as it is only a proxy.