So this isn't really as bad as it may seem. In my opinion, the hardest part of this was round four, after that, it's not too bad if you buy the right perks.
Arcade mode is located under singleplayer on the main menu.
To beat arcade mode you need to make it through 8 rounds with 5 lives.
After each round, you can buy perks and powerups that will make your life easier and you can also buy an extra life I believe up to three times, however, I never needed more than three.
Round 1-3 are very simple to get through, just throw the ball at the opponent and try to catch it when they toss it at you. I believe after round 2 the powerup to increase the catch radius on the ball the opponent throws appears, so I suggest you buy that as soon as you see it because it will make catching the opponent's balls a lot easier.
Round 4 can be tricky because of the boss battle at the end, the laser bot can throw balls at you extremely fast so you want to constantly be moving. I don't know how the damage works exactly, but I would sometimes throw the ball at the boss and take away 3 quarters of his health, but I would also hit him and it would only do 10 damage.
After the fourth round, you have a chance to buy the homing powerup. You want to but this IMMEDIATELY. It will make your life so much easier for the last 4 rounds. If it doesn't appear after the 4th round, it almost always appeared to me by the end of the 5th round, so don't give up hope. Once you get the homing powerup you can throw the ball at any bot in your sights and the ball will almost always hit them unless it gets blocked. This makes the higher rounds much easier as they start sending upwards of 6 bots at you at a time.
So just continue with the homing until you come to the final boss at the end of the 8th round. This boss is not that hard as long as you get good at catching balls, and dodging. The only way you can do damage to him is by catching one of his balls and then throwing a ball at him, then repeating that process. When he throws a ball, look to see if it has orange flames around it before trying to catch it, because you can't catch the flaming balls. Just look for one without flames and then try and catch it, which should be significantly easier with the catching powerup.
This is only my 3rd guide ever on this site so if you have any suggestions please let me know. I saw that this achievement didn't have a guide, and it was also the least unlocked achievement in this game. So if you are reading this and were thinking about starting this game, just know that it's really not that bad. Just keep at it until you can beat round 4 or 5 and get the homing perk.