The verbs built according to the model
на-<verb>-ся mean "to do the action so much that you do not want to do it any more, to do the action so much that you feel satisfied with the result". All the verbs ending in
-ся, the reflexive verbs, are intransitive, they cannot have direct objects, that is, they cannot have an Accusative case object. In the case of the
на-<verb>-ся verbs, the object with which the satisfaction is achieved is usually in the Genitive case.
Typical examples of such verbs are наесться 'to eat until you are fed up' and напиться 'to drink that much that you are not thirsty any more'. In the following examples the Genitive objects are italicized.
Вова наелся варенья. - Vova ate jam (until he was fed up with it and didn't want to eat it any more).
Колодец, колодец, дай неба напиться! - O well, o well, give me some sky to drink (so that my thirst ceases). - From a song.
Гости на свадьбе напились самогонки и танцевали до утра. - The guests at the wedding party drank so much moonshine that they danced until morning.
Я не видел Олега 5 лет, просидели весь день, никак не могли наговориться. - I haven't seen Oleg for 5 years, we spent the whole day together and couldn't stop talking (however much we talked it never seemed enough for us).
As for your two sentences, the first one is right. In the second one, the object should be in the Genitive case, and usually after начитаться you simply say the name of the author, because it is clear that the writer writes books, so it should be Он начитался Пушкина - "He read Pushkin much" or "He read Pushkin so much that he doesn't want to read him anymore".
Он начитался Пушкина и сам начал писать стихи. - He read Pushkin much and began to write poems himself.
Он начитался Троцкого и вступил в коммунистическую партию Парагвая. - He read many books by Trotsky and joined the Communist Party of Paraguay.
Иди спать, ты что, ещё не начитался своих комиксов? - Go to bed. Haven't you read your comic books enough yet?
From the verb начитаться the adjective начитанный "lettered, well-read, deep-read" is formed.