Gyokeres likely won't have too much trouble adapting, as he's already quite familiar with the Premier League, having previously played for Swansea City and Coventry City on loan from Brighton. I think Gyokeres will be able to perform as expected, befitting his very high price tag. While he won't immediately become the Premier League top scorer, with the depth of Arsenal squad, I think Gyokeres will be able to score more than 15 goals in his first season.
Arsenal chances of winning a trophy next season will automatically increase, as Arteta now possesses all the necessary attributes to make that happen. However, I still think Liverpool and Manchester City have slightly better chances than Arsenal. Overall, Arsenal will be a team to be reckoned with in the race for trophies, and if they fail again, perhaps the club top brass will sack Arteta at the end of the season.
Yes he will adapt quickly because he has played in English football before even though it was nog directly in Premier League. Given his impressive 68 goals in 66 matches for Sporting CP and his high transfer fee you expect him to perform well. And yes he is not top scorer right away I also think he will score over 15 goals in his first season due to Arsenal strong squad. Arsenal chances of winning trophy will greatly improve with coach Mikel Arteta now having more complete team but Liverpool and Manchester City slightly higher as compare to them. But truth is that Arsenal will be serious contender for trophies.