The main thing Akiha has is her absorption through all of her abilities. Basically, if her hair, or any abilities of hers, touches you, it drains your heat and your life energy. It can, in fact, be a projectile. See
here,
here,
here,
here,
here,
and here. In character, she will usually start with the last few, and not really use the first 2 very often. and seeing as this is the Complete Inversion key, she will try to kill at the first opportunity.
Interesting, I don't think The Terrarian has heat draining, but I should mention that the Terrarian can also slowly drain the life of his enemies as long as they're in his range using his Life Drain
https://terraria.wiki.gg/wiki/Life_Drain. It's not an instant kill, it's not touch based nor projectile based, but rather it's an ability that functions towards everything that's in his range. It can even function through walls.
Though I should mention that it can be unlikely that he uses this weapon since he has weapons that deal a lot more DPS and this weapon is a mid-game weapon compared to his other endgame weapon, I am just saying it's an option.
That being said, The Terrarian can quickly get out of her melee range and the range of her projectiles using his Rod of Harmony. It's safe to say that in equal stats, a character that can fly, teleport and attack from a distance has more agility than a character that lacks those abilities.
Addressing the actual place where the fight is, the back rooms. In any normal situation, Akiha would actually have a decent advantage, as she actually has experience fighting in enclosed areas. but:
I agree, Akiha being in an environment that is close gives her an advantage, but that doesn't necessarily mean The Terrarian doesn't have experience fighting in enclosed areas. The Terrarian has multiple weapons that are close ranged like the Starlight
https://terraria.wiki.gg/wiki/Starlight for example. So he definitely isn't a slouch here, it just means that he will have difficulties using his flight advantage.
I would just say "there's nothing stopping Akiha from destroying the wall", which would be true. She's completely insane at this point in time and doesn't know restraint in the slightest. She would try to destroy everything in sight. The only reason she didn't in the VN was because she could very easily see and keep track of where Shiki was very easily as he was constantly in her range and is also a close range fighter.
Fair, but I should mention that even if Akiha can destroy walls, The Terrarian should still be an advantage. Since she can't see where The Terrarian hides, she would be attacking randomly and guessing where he should be at all times. While that's happening, The Terrarian could distract her for enough time to overwhelm her with his enormous amount of attacks. He can literally summon several magic flying swords that attack her with the Terraprisma
https://terraria.wiki.gg/wiki/Terraprisma while he makes a rain of wall-phasing projectiles rain all over her.
Since the Backrooms is a place with an endless amount of walls, The Terrarian could just teleport behind other walls if Akiha destroys the walls he was hiding in and continue the game of cat and mouse where Akiha trying to find where The Terrarian is hiding is while he is overwhelming him with multiple minions, sentries and projectiles.
wouldn't this be more of in their sight, since it's only where the cursor is? I don't play terreria.
no walls, no problem
Well it's a weird thing, Terraria is not a normal first person game, it plays more like a 2D platformer. That way, The Terrarian can see things through walls. Inside the gameplay, he can hide inside buildings, holes and walls and attack enemies with weapons that phase through walls like the Solar Eruption
https://terraria.wiki.gg/wiki/Solar_Eruption for example, which is a flail-whip-sword thingy that attacks through walls. Terraria characters being able to see through walls is consistent with the fact that no matter if you hide inside the floor, your enemies and bosses can still see where you are. The Queen Slime and King Slime for example will teleport on top of you if you try hiding inside a building where first person 3D characters normally wouldn't be able to see you at.
being able to fly here won't be of much help.
for the love of god, I hope so
I'd still say flying would help The Terrarian a lot, even in closed environments. He can fly infinitely towards any direction using wings or mounts like the Cosmic Car Key
https://terraria.wiki.gg/wiki/Cosmic_Car_Key, which would give him a lot more maneuverability throughout the terrain while fighting Akiha.
I would also like to mention that depending on how the terrain works, The Terrarian can blow up the environment with his weapons.
He can make a hole in the ceiling and fly upwards to make more room where he can dodge better using his infinite flight, or he can blow up the floor under Akiha making a deep hole where he can attack her with his superior range while Akiha would struggle to get to him.
He can easily do this with the Celebration MK2
https://terraria.wiki.gg/wiki/Celebration_Mk2 which shoots dozens of missiles in quick succession with Mini Nuke II's as projectiles
https://terraria.wiki.gg/wiki/Mini_Nuke_II which would let him shoot dozens of missiles that can destroy the environment and alter it to The Terrarian's advantage.