The Gunners were outplayed by Everton in their 3-0 defeat at Goodison Park, putting their Champions League place under threat with the Toffees just one point behind with a match in hand.
Everton will overtake them in fourth with a win at Sunderland this weekend, but Arsenal's attention will be on the cup this week.
Ramsey, who came off the bench for his first appearance since Boxing Day after a thigh injury, said the priority was to improve immediately if they were to end their nine-year wait for a trophy and then they could re-focus on their league campaign.
"We have to pick ourselves back up now," he said.
"We have still got a lot of points to play for. We are in the semi-final of the FA Cup, so we have got to put things right.
"We have to put a strong run-in together now. We have five games left and we will aim to win them all to give us the best opportunity to finish fourth or higher.
"I think Everton still have a few difficult games but, then again, every game is difficult in this league.
"So many teams can upset each other so we will have to be aware of that, concentrate on ourselves and just play for the remaining 15 points."
The only positive note from the defeat on Merseyside was the long-awaited return of Ramsey, who was a key player for Arsenal before his injury.
He is now looking to make up for lost time by securing some silverware and the club's 17th successive season in the Champions League.
"It was very frustrating for me to be out for three months of the season when things were looking so good," he added on arsenal.com.
"These things happen in football and I am just glad to be back out there now.
"Hopefully I can show what I can do, help this team achieve Champions League football again and hopefully have a medal at the end of the year."
Manager Arsene Wenger hopes taking on SkyBet Championship side Wigan this weekend in the cup will give his team a chance to bounce back.
"Once you are in the semi-final you want to go through to the final," he said.
"To play in a different competition will help us find confidence back and focus on our next game.
"We have to acknowledge that we must go back to basics and come in much stronger in the challenges, much stronger in the way we win the ball back, before we start winning games.
"At the moment I feel it is important to focus on the quality of our game before we dream of where we finish (in the league).
"We have to come back to much better quality than we produced at Everton."