「それはわしがやりおまえが追いつめられたのでわしの攻撃と同じ事!」
Allow me to first insert a couple of commas for easier reading.
「それはわしがやり、おまえが追{お}いつめられたので、わしの攻撃{こうげき}と同{おな}じ事{こと}!」
This is a 100% informal spoken line; therefore, a serious analysis of its grammatical construction may or may not prove very productive. So, I will try to keep it light.
「やり」 is the 連用形{れんようけい} ("continuative form") of the verb 「やる」 meaning "to do". As you can see, the sentence still continues after the 「やり」.
「やり」, in this sentence, is tense-less as far as its physical form (as there is no 連用形 for verbs in past tense to begin with), but the action certainly took place in the past. Thus, it means "やった and then" here, and that in turn means the same thing as 「やって」.
Cause: わしがそれをやった "I did it", "I am the one who did it."
Effect: おまえが追いつめられた "You were up against the wall."
Conclusion (in speaker's opinion): わしの攻撃と同じ事 "It is the same thing as my attack", "I should get the credit as my attack."
Hope this makes sense. If this is from a game, I know nothing about games.