“Not that this daybreak would be up to much even when it finally got rolling; call it dawn with a hangover”
(Note that "even" is not part of the idiom.)
Let's analyse each part.
not up to much (idiom)
to not be of good quality
to get started
With this King is referring to the progression of sunrise (at the very beginning vs half way through).
Consider the symptoms of a hangover - you generally feel terrible, aren't up to being your usual self, look pale and sickly (as Edwin Ashworth notes).
So this sentence basically means that the sunrise wasn't very good even when it got properly started - maybe it was just a plain yellow instead of a gorgeous mix of pink and orange, and the colour was washed out and barely there instead of being vibrant.
The amusing personification with the "hangover" should hopefully make sense now. The dawn is not up to its usual standards, so if you imagine it as a person, their condition could be explained by them having a hangover.