Is there anything special about the word "the day-before"?

The day-before is explained in the text—the grandmother says it as a single word.

“Him” means “it”. The grandmother is declining this pronoun for gender, so apparently a turkey is masculine in Italian (at least this turkey).

These are both attempts by the author at dialect—an attempt to show that the grandmother speaks English as a second language, which might be called “broken” rather than bad; she knows some English but does not speak idiomatically.

Often attempts to convey dialect are not very accurate or are partial, just enough for the reader to get the idea that the person is a non-native, non-fluent speaker of English, perhaps someone who immigrated in middle age.

I would not attribute anything beyond that to the dialect. The brief paragraph is not enough to tell whether the grandmother is an adroit or sophisticated speaker of Italian.

The book overall is an LGBT young adult love story about two young girls with what is reported to be a happy ending.