The Battle
Refer to the Phantom 1 strategy: everything written there is applicable to Phantom 2. The only real differences for this battle are the arena (there are no pillars for you or Phantom to hide behind any longer) and Phantom's new attack, which can be cancelled with Stingers to his face before the fireballs come raining down.
Here is an alternative strategy you can use to “win” the battle by default :p
Ceiling Glass trick:
If Phantom lands on the ceiling glass 5 times, the battle will automatically end – you will know if you’ve got a ‘hit’ which counts if shards of glass fly up when Phantom lands. At the beginning of battle, run straight to the centre of the glass ceiling to trigger Phantom into leaping after you, wait one second or so, then jump away as Phantom descends. From then on, you will only need 4 more landings. To trigger Phantom into doing a jump, get right behind him – I mean “literally-touching-his-rear-end” (sorry >_>) behind him. Also, taunting while you’re behind him seems to often provoke Phantom into doing a jump as well. Once Phantom jumps, run to the glass ceiling to make sure that Phantom lands on there.
Note 1: If you use this method, then you won’t get DT restored after battle: so if you want full DT after battle, taunt or attack Phantom during battle to restore DT – damaging him won’t affect the amount of landings needed to end the battle.
Note 2: On Hard/DMD mode, it still only takes 5 landings for Phantom to crash through the glass.