Yes, at least for the neurotransmitters that a given cell releases.
As far as the neurotransmitters a cell responds to, you are really better off thinking in terms of receptor expression. Most neurons will express receptors for many different neurotransmitters. It's far too broad to go into them all in an answer here.
As far as the transmitters that cells release, morphological types of neurons are very often associated with release of particular neurotransmitters. In the cortex, for example, pyramidal cells principally release glutamate while there are potentially dozens of morphological types (again, depending on how you define them) that release GABA; in the cerebellum, Purkinje neurons release GABA whereas granule cells release glutamate.
However, be careful: some terms like "granule cell" don't really refer to a special morphological type. They are really just describing "small cells in close proximity": these can be completely different types in different brain regions. It really makes no sense to ask these types of questions for the brain as a whole, you have to study individual brain regions, each of which is organized quite differently from the others.