The Battle
This is a pretty tough battle for a beginner to the game, but - never fear - this not entirely chummy scorpion/spider hybrid can be defeated :p
Probably the easiest way is via Air Raid - buy a purple orb if you can afford it, too. (If you can't afford even that much, you could always replay Missions 1 and 2 and orb-farm until you can, I guess.) To activate Air Raid, jump, press L1 to DT, R1 to go into flight mode, then mash on the shoot button to zap Phantom. If you fly behind him, it's sometimes possible to hit him in two spots simultaneously - head and back - for double damage. When this happens, the life just falls off him.
With this strategy, you'll be running out of DT frequently. You can regain it in a number of ways: taunting, melee attacks, firepower. To taunt, press and hold down R2 whilst near Phantom - always be ready to cancel out of a taunt by jumping away should Phantom start up an attack in the meantime, though! Otherwise, just keep on attacking Phantom's body with downward slashes (it doesn't matter if you clang against armour; some DT will still be restored) or walking around, firing E&I or the shotgun.
Phantom has two weak spots: his mouth and his rear end - or, to be more precise, the base of his tail/thickest part of his bulb depending on whether he is in scorpion or spider form. Air Raid is one way of hitting them, but there are other ways too.
One such way is downward slashes (aka Helm Breakers in DMC3, if you're familiar with DMC1's prequel). I wouldn't do downward slashes to Phantom's mouth - not unless you have Air Hike, at any rate - but you can do them to Phantom's rear end. To give you an idea of their power, just 4 DTed Air-Hiked downward slashes can finish Normal Phantom off if they land on the right spot every time. You can also do downwards slashes to Phantom's back for lesser damage.
If this is a new game, you'll be extremely unlikely to have Air Hike yet, but no matter - single jump slashes will do fine. If you're feeling ambitious, you can try going for enemy-hiked slashes: do a short jump towards Phantom's rear end, kick-jump off it, and slash downwards for equal damage to an Air-Hiked slash. It's pretty tricky to connect with Phantom's rear end to hike off it, though.
Anyway, you get the idea. Just keep on jumping and slashing away at Phantom, and DT whenever you have it.
A second such way is Stinger, preferably Level 2. (Though Level 1 can be used at a pinch.) Stinger is used to attack Phantom's mouth. If you spot Phantom charging up a fireball or dipping his head into the ground to summon pillars of lava, you can interrupt the attacks with a Stinger to Phantom's head. A Stinger will normally make Phantom flinch, i.e. roar and rear up in pain. You can use this to trap Phantom in a simple Stinger loop, which goes like this:
Run up to Phantom's face to trigger his slow swipe with both claws > jump backwards > Stinger his face > wait around in front of his face to trigger another slow swipe > rinse and repeat.
Once you get that pattern down, you can try sneaking in a second, less-damaging, Stinger. After the first Stinger Phantom will flinch. Wait on the spot for him to land back on the ground, then Stinger him - jump away quickly, since he won't flinch this time! When Phantom's life goes into the red, you should abandon the second Stinger, but otherwise you can still use the Stinger loop in his red phase.
Offense-wise, that's pretty much it for Phantom: Air Raid, downward slashes, Stingers. If this isn't a new game and you have the Nightmare Beta, DTed Nightmare-Beta shots work surprisingly well against Phantom since the shots penetrate through his armour. You could also use Ifrit, but Alastor so much has the edge that it generally isn't worth it - fully-charged Meteor Lvl 2 shots do a ton of damage, though, if you can hit Phantom's face - a big "if".
Defense-wise, you can get behind a pillar to avoid Phantom's fireballs, but that's about it. You're far better off fighting Phantom in the central aisle, where you have space to move and can (usually) see what's going on. It's best to avoid going into the corners or the low-lying corridors if you can - only go into the corridors to flush Phantom out, should he retreat there to fire off fireball after fireball. If Phantom disappears out of view, then hold down R1 to make the camera (slowly) pan around behind you and bring Phantom into view - he gives audio cues for all his attacks so you can still evade him even if you can't see him.