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.
In the Premier League it might not be easy for him to score 68 goals in one season because it is a much tougher league than the Portuguese league.
He can adapt quickly because he has the ability to be a top player, but performing at his best is not easy and takes time.
Arsenal already has a strong squad next season, and now their chances of chasing trophies are very good.
If after acquiring quality players, they still fail to win the title, then I think Arsenal are simply not ready to compete for the title, and this season is a great opportunity for Arteta to prove that.