I am writing this merely for a reference for those wishing to chase this achievement. Honestly it can be very simple to get, or it can be very difficult depending on how active your server is with PVP participation.
A few things you need to know first:
All references should be checked with wiki.bluegartr.com or ffxiclopedia.org, both have lots of info, and easy to follow guides on things I will be referencing below.
1) You must have your Ballista License Key Item to participate
2) You must have completed up to a certain point in the story missions for your home nation (San d'Oria, Windurst, Bastok) to begin the short quest to gain the Ballista License.
3) To follow this guide, it will not matter which job you are playing as, or if your combat/magic skills are capped (I mention this due to the addition of Abyssea leveling parties for those that are not aware or haven't been played the game in the last two years)
4) Please understand that Ballista is on a rotating schedule, it will move between Jugner Forest, Merphitaud Mountains (spellcheck) and Pashow Marshlands, and each event is limited to players that are of the allegiance for the upcoming match, i.e. Bastok vs Windurst. If you are not sure which zone/time your match should take place, please use the above mentioned references.
Now to the actual guide.
You have two options depending on how to go about this, first you can gather 5 friends in addition to yourself, in game, for a 3 on 3 setup (need to have a minimum of 6 players signed up before the match or it will not start), and all players must be from the nations that are named for the upcoming match. Second option, if you have multiple accounts, meaning you have several playonline id's and can dual/trio box, you can run this all by yourself if you have 6 alternate accounts total or any combination there of (6 different players, 2 players with 3 accounts each, 3 players with 2 accounts each, etc.)
Once you have gathered enough players to participate and you are ready for the next match, go to the zone in which the match will take place, find the Herald (its a npc, in the zone and it will only appear while registration is taking place, so don't panic if you go looking and its not there) have everyone sign up for the match, and be in the zone when the match starts. It costs a small amount of gil. The match will start and you get a short cs with the players lined up. AFTER the match starts, the player attempting the achievement needs to run around and Quarry for petras, after getting a few run them to a Rook and score points. The other players don't have to stay as the match takes an hour before it will end. To be on the safe side I had one player on the opposing team remain.
After the match concludes you will check with the npc and ask them for your results/reward, it will give you a message that you will need to wait a short time for this to happen. As long as you were the top scoring player, you will get the option for two Temporary Key Items, One being the top rank for the job you were playing as, do not choose this if you want the achievement, second it will let you choose the Gold Ballistar Temporary Key Item, this will only stay with you for one week in real time.
After gaining the Key Item, go to a town and visit a bard while playing on your Xbox 360, and unlock the achievement.
If you are on a server that has an active Ballista Linkshell, meaning there is a group or groups that do the event on a regular basis, this will be much harder, as you have to place first overall for the zone based on the last 20 matches, to see if there is anyone actively doing Ballitsa, visit Ru'lude Gardens in Jeuno and check with the npc at the top of the stairs leading from the Mog House, it will show you scores from previous matches for each zone. If they are blank then no one is doing this actively and youre good to go. If you find they are not blank, I would suggest trying to speak with the group directly and inform them what you are attempting to do, they may be willing to help you.
If you have any questions feel free to message me on TA, on Xbox Live, on FFXIAH.com, or ingame if youre on my server. My Character name is Raist, and I play on the Phoenix server. Im more than willing to help with most anything you may need.