The Colombia international joined Arsenal from Nice earlier this summer, and the Poland international is looking forward to battling with the new signing for the No. 1 jersey.
Ospina completed his switch to Emirates Stadium from Ligue 1 side Nice last month after impressing for Colombia at the World Cup.
"I love a challenge," he told Arsenal's official website. "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."
Arsenal also has added Alexis Sanchez, Mathieu Debuchy and Calum Chambers to its ranks in the offseason, and Szczesny believes the club can mount a challenge for the Premier League title in 2014-15.
"We were close to being right near the top last year in the Premier League and we managed to win the FA Cup," he added.
"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."