Unlike Pinball FX2, you only have to defeat all the spiders, there is no time limit. You do not even have to beat the mode (although if you've already hit all the spiders you have essentially finished anyway- you just have to hit the exit ramp. I got this without hitting the exit ramp).
To start, you need to begin the Arachnid Attack main mode. Hit all 3 of the flashing circular targets at the top of the table (no need to hit each individually, hitting any of the 3 will light up another one instead of needing to hit each specific one in FX2). Once this is done, make sure Arachnid Attack is flashing (you can change which one is flashing by hitting the bumpers on the left side. You just have to get a bit lucky. You could also just play through all modes until you get to the one you want). Now hit the flashing cellar sinkhole on the right side of the circlular stairway that raises up.
Once in the cellar, hit all 5 of the spiders twice. Each time you hit a spider, a second one replaces it after a few seconds. Once you have hit all 10 spiders, the trophy unlocks. You can choose to exit through the ramp and complete the mode or just let the ball drain.