If I activate Ravenborn Tome, does the buff of 1 bonus damage stay after the end of my turn?

No, the effect from Ravenborn Tome does not stay after the end of your turn.

According to the card's text, it is a delayed ability that specifically applies to "the next spell you play this turn." Furthermore, per Rule 317.2.c, all "this turn" effects expire simultaneously during the Expiration Step of the Ending Phase. Therefore, if the effect was not consumed by playing a spell during your turn, it will expire when your turn ends.