Runarsson joined the Gunners in the summer from French club Dijon in a deal worth around £1million.
The 25-year-old has slowly been adapting to life in England and he has only appeared for Arsenal in the Europa League so far.
Runarsson is expected to play again in the competition on Thursday when Arsenal host Rapid Vienna at Emirates Stadium, with Leno rested.
The match is another opportunity for the Iceland international to shine and he is eager to challenge Leno to become Arsenal’s No1.
And asked if he backed his own ability in the fight with Leno to be first choice, Runarsson said: “Yes, 100 per cent, otherwise I wouldn’t be playing for this club, I think.
“I have to have this mentality that I can play and I have to believe it and show it.
“I think that should be the mindset for everybody, that you want to play. It doesn’t matter what position you play.
“I think maybe as a goalkeeper it is a little bit different, but you definitely have to have the mentality that you want to push to become No1.
“If it happens or not, if it happens in one week, one month or one year, that always has to be your mentality - because otherwise you’re in the wrong profession I think.”
Arsenal swooped for Runarsson after selling fellow goalkeeper Emi Martinez to Aston Villa in the summer for £20m.
Runarsson came highly recommended by the Gunners’ goalkeeping coach Inaki Cana, who worked with him at Danish club FC Nordsjælland, and he has been helping him settle in England.
“Our styles are very similar, so for me personally he can help me improve on the things I need to improve,” said Runarsson.
“He is a specialist in those things. I think, in general, he is a little bit extreme, very demanding, but in a good sense.
“He is always after you. There is no time to relax, and that is what you need if you want to improve.
“It is good when you are settling in to know somebody, because that is going to help you adapt more quickly. He has helped me tremendously to settle in more quickly.”