The crux of this fight (PvP), compared to any other fight (PvE) in this game, is that Apollo is as quick on his feet as Lea should be, maybe even better. His raison d'être is to make sure you do, or at least know how to, effectively use a group of core game mechanics, called Combat Arts. He'll keep you on your toes with the sheer amount of Arts he'll use on you. If you hope to have a chance of beating him without losing a single round (whether for the Trophy/achievement or just to prove your grit), you'll have to use at least a few yourself, which means levelling up and spending those Cross Points on your Circuits - but not completely randomly, you do have to choose the perks you plan to benefit from.

If you're reading this you probably wanna get it right the first time. Remember to save before each fight, you'll be reloading it quite often if you're anywhere as bad as I am.

• Remember that there are 4 types of Combat Arts, which each branch in a group in one direction on the skill tree: Melee Arts, Throwing Arts, Dash Arts, and Guard Arts.
  
  
• You can activate one by hitting the Combat Art key (default bind is Space) while performing the relevant action (i.e. hit Space during melee swings to activate a Melee Art, or during a dash to activate a Dash Art).
  
  
• Holding the key should power up the level/strength of the Art you use, depending on whether you unlocked it yet and if you have enough Special Points (the units on the meter next to your health bar).
  
  
• In general, I would recommend prioritising Melee Arts, Guard Arts, and defensive passive skills when grinding Cross Points for upcoming fights with Apollo.
  
  
• During Apollo's vulnerable phase (during which he'll be flashing red and moving sluggishly), spending a Special Point or 2 to use a Melee Art on him would help cause a lot of extra damage.
  
  
• In case you can't dash-dodge all of his balls (he's very fast), remember to raise your Guard to absorb the few inevitable projectile hits. When he comes in close for a melee attack, activate your Guard Art right before he lands a hit (or at least before he lands a second hit).
  
  
• I don't recall ranged attacks or Throwing Arts ever being very useful in any of these battles, except when Apollo was vulnerable and I was too far away to use a Melee Art on him. I don't recommend using Throwing Arts at all, and only using ranged attacks during vulnerable phases, if ever. Apollo is too quick, and easily dodges nearly all ranged attacks. Shooting slows down your own movement, and you need to be as fast as Apollo if you want to dodge his attacks.
  
  
• This tip is entirely optional, but if you can master Perfect Guarding (raising your guard in the instant before an enemy lands a hit on you), it would stun Apollo and allow more opportunities to whittle his health bar down. If this sounds appealing, it's worth the practice as it can come in very handy.
  
  
• Remember to practise your new skills on local monsters before fighting Apollo, so you know how to use them effectively.

When Apollo initially challenges you, there will be an entire area (Autumn's Rise) to grind and complete quests before you reach the fight's location. You should easily be able to level up since the early levels don't require so much XP, and spend the CP you earn from levelling up on Combat Arts. Spending any CP on Throwing or Dash Arts so early in the game would be inadvisable, since enemies are still easy enough to deal with using mainly Melee, Apollo is hard to hit with ranged attacks, and he doesn't take enough damage from Dash Arts to be worth unlocking those yet.

At this point, neither you nor Apollo will have access to much overpowered gear or any damage types outside Normal, so just equipping the gear with the highest levels or best defense stats that you own should be fine. (If you want to take absolutely every precaution, grind for the Bronze set, sans the Goggles - get a Cooling Veil instead.)

This one gets a little more challenging, as by this point you and Apollo will be playing with fire. Remember not to spam elemental attacks, to avoid Elemental Overload, since you want to be able to use Heat attacks when Apollo's vulnerable (they do more damage than the base damage type, Neutral).

Dash Arts will be more useful this time, as well as some Heat-resisting passive skills, so unlocking those on your skill tree would be smart. As far as I know, using Heat on another "player" shouldn't have resistance, unlike using Heat on Heat-type monsters, so feel free to spend the entire fight with Heat mode activated (except when your Elemental meter is close to filling, obviously).

Don't stay too close to Apollo, he can easily burn you with his new Throw Arts. He comes in close when he intends to melee and you can more easily Dash Art through him and Melee Art whenever he's vulnerable (red) or stunned by a Guard Art, but if you can't K.O. him during the round where he places Heat landmines before he actually places the mines, then dance carefully around the perimeter of the ring - they cause massive damage to you if you slip within their radius.

(If you want to take no chances with gear, I recommend the Steel set.)

By this point, you hopefully know how to deal with most of Apollo's tactics - just stay out of his reach but activate Guard Arts when he comes in close for Melee. The only thing to note this time that I can remember is that if he switches to an element, make sure you're not currently in the inverse of that damage type, as that would leave you susceptible to extra damage. (E.g. if he's using Heat attacks, make sure you're not in Cold mode, and vice versa.)

With skill, you should beat Grasshead easily. Whether or not you succeed, I appreciate feedback, even if my guide sucks. Good luck!