The Russians were beaten 3-0 by eventual winners Spain four years ago, and Arshavin is keen to reach the final four of the competition once more.
"Getting to the semi-finals would be a success. because then we won't have lowered the bar from last time; and also because to get there we will have to beat one of the best teams; Germany, Netherlands or Portugal," Arshavin stated to Sport Express.
"Spain were simply stronger than us [in 2008]. Not only stronger than us, but than all the other teams in the tournament. We held them for 70 minutes but when you just do that; chase the ball against a machine like that, you just run out of steam."
Russia recently beat Italy 3-0 in a pre-tournament friendly, but the result did not come as a surprise to Arshavin and his team-mates.
"We've known for a long time that we're capable of playing at that level. Most of the players started to feel confident in their ability when first CSKA and then Zenit won the Uefa Cup.
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"We all know very well how to prepare and get that sort of result. Plus, [Guus] Hiddink from his first days with the team brought us conviction that there is no difference between us and the other Europeans, we are one and the same."
Russia start Euro 2012 with the Group A match against Czech Republic on June 8.