To me, the Arsenal team were not that desperate for the cup after many points ahead of their close rival blinded them. The morale was so low, but little did they realize that it was not over until the end of the season.
Having no doubt is quite optimistic. All the Premier League clubs are swimming in money. Finishing top 4 is nowhere as hard as it is in the Premier League. There is Manchester City, Liverpool, Manchester United, a Newcastle team that could invest more than anyone else, but also Chelsea and Tottenham who will both do everything they can to change certain things and get going again for next season. Chelsea in particular will attack next year to get back into the Champions League. Arsenall can finish in the top 4, but it isn't easy at all. I doubt that Liverpool will be as weak again once more.
Arsenal also has a harder year in front of them than they had this year since they are also playing the Champions League. It is not the number of games, but the mental burden is higher when you want to be successful in that cup.