To gain access to the Dragon cup, you need to battle ordinary fights in the Arena, which is located in the city Liberdam.
When you're looking at the available fights list, there'll be a bar at the top, with crowns.
Those crowns at the cups, the fourth cup is the Dragon cup, so win enough battles to get up to that cup then you're able to participate.
The fight itself is level 60, consists of a singular dragon.
If you have the item on you named Igzakus' Necklace, it'll help TREMENDOUSLY.
Due to the following stats it gives.
Igzakus' Necklace
Causes damage to beasts +50%
Causes damage to dragons +200%
Causes damage to trolls and other giants +200%
I was able to complete this battle with that necklace, with Areus being 49, Selene 48, Dagda 47.
The tactic I used was stand at a distance and just shoot the dragon with Magic Arrow.
When your Tri-Attack is full, go in and use it on the dragon, then just spam your basic attack on it as much as possible, when the ring is about to go away, quickly use Dark Meteor. If done right, your Tri-Attack bar will be full again due to the amount of hits you dished out, so once again use it ASAP, because if you're quick enough, Dark Meteor will still be in action, so that combined with your swords will stun the dragon, giving you a longer period to attack and weaken it.
Rinse and repeat this tactic and you should be all fine and do this in a minute, or two.