This is the 264th most frequent Ukrainian word.
наступного
"Next" or "following."
In this sentence, 'наступного' is used as an adjective in the genitive case to indicate something related to the following day.
Я зроблю це наступного дня.
I will do this the next day.
Here, 'наступного' qualifies 'разу' (time) to denote the next occurrence.
Ми зустрінемось наступного разу.
We will meet next time.
In this context, 'наступного' refers to the subsequent letter, modifying the noun.