I will try to provide some geometric intuition, why there should be an analogy between local rings and graded rings with unique homogeneous maximal ideal. Maybe this also helps to guess whether a statement true for local rings should still hold in the graded case.
A graded k-Algebra can be thought of as an affine space with $k^*$-action. Homogeneous prime ideals correspond to invariant closed sub-varieties. So your sort of algebras corresponds to spaces with exactly one fixpoint. For example in the case of the polynomial ring its $k^n$ with the obvious action of the multiplicative group and $0$ is the fixpoint. From this example we see, that the action can be used to "contract" the space to the fixed point. Hence the local nature of the space.