Ospina impressed for Colombia at the World Cup in Brazil and agreed to move to Emirates Stadium last month.
The 25-year-old replaces Lukasz Fabianski, who left Arsenal for Swansea City at the end of his contract, and Szczesny is ready to embark on another friendly rivalry with Ospina.
"I love a challenge," he told Arsenal Player.
The confidence is flying high and hopefully this year we can challenge for the Premier League
"I was expecting someone to come because Lukasz has left for Swansea and I wish him all the best. I knew someone would come in. I'm just going to do the same thing over again, I'm going to try my best in every game and every week in training. Hopefully that will be enough."
Szczesny started Sunday's Emirates Cup defeat to Monaco and he believes Arsenal's fitness levels are close to where they need to be ahead of the new season.
"It would have been nice to win a cup that we play at home, but that's not what we talk about in pre-season," he said. "We want to prepare and make sure we are at our very best at the weekend [in the Community Shield] and then against Crystal Palace in [the Premier League].
"Last year we had a long pre-season and we played a lot of games in Asia. This year is completely different but I think everyone is looking sharper every single day.
"We were close to being right near the top last year in the Premier League and we managed to win the FA Cup. This year we added some players and we're looking stronger. The confidence is flying high and hopefully this year we can challenge for the Premier League."Copyright 2016 The Arsenal Football Club plc. Permission to use quotations from this article is granted subject to appropriate credit being given to www.arsenal.com as the source