a. He aspires to little higher praise than being called a good writer.
a1. He doesn't aspires to much higher praise than being called a good writer.
(Being called a good writer is basically enough for him and he doesn't need much higher praise.)
b. He is as good as his father, and there's little higher praise than that.
b1. He is as good as his father, and there isn't much higher praise than that.
(He's as good as his father, and there is not a lot of higher praise than that.)
Are the sentences grammatically correct and do they correspond the the meaning sentences in each case?
In the first two sentences 'little' and 'not much' modify 'higher', but in the second two sentences they modify the noun phrase 'higher praise' and are about the amount of higher praise possible.
Maybe in the first two, one could use 'praise little higher than'...