Achilles doesn't need armour. Minus his heel he's invulnerable. So Punisher can plug away until he runs out of bullets.
This would end up a close quarters fight and while Punisher may be very good in that regard Achilles is better as that's his style. Even if you want to take away his weapons Ancient Greek warriors knew grappling and wrestling so Achilles could get Punisher down and snap his neck
Achilles doesn't need armor? That's weird, because he sure wore a lot of it...
What do you think is going to happen to Achilles when Punisher is shooting him? He might be bulletproof, but that doesn't mean the laws of physics don't apply to him. The force of the bullets would likely knock him on the ground, or at the very least make him unable to advance. I don't think this fight would ever get to the point where it becomes hand to hand. Punisher would use bullets to pin Achilles down, toss in a grenade, and that's the end: the explosion would easily rip Achilles' heel out, thus killing him.
Even if it did come to a close combat fight, I still think the fight easily swings in favor of Punisher. Greeks may know how to grapple, but Punisher is also an immensely talented hand to hand combat, and is an master practitioner of Nash Ryu Jujutsu, Ninjutsu, Shorin-ryu Karate, Hwa Rang Do, and Chin Na, as well as Krav Maga knife combat.
Kevlar wasn't meant to hold up against piercing blows. Achilles had a sword in The Iliad. It's also worth noting that whereas Achilles would be wearing a helmet, Punisher doesn't, leaving more exposure for an attack.
Punisher has been fighting bad guys with guns for a long time and has only gotten shot in the head a handful of times. If he's able to avoid getting shot in the head by bullets, he'll surely be able to avoid arrows and javelins.
To your point regarding Achilles's sword... I don't think he'd be able to get in range to even use it. Punisher would use guns and grenades to keep him at bay until the opportunity to shoot Achilles in the heel presented itself.
Achilles is a Greek hero with nymph blood. Punisher is a human. How is Punisher going to overpower the same guy that the gods had to forcibly restrain from sacking Troy? It's not about who knows more fighting techniques. It's who could execute them better. Achilles would beat Punisher old school.
Yeah, the gods restrained him from sacking Troy, but not from sacking Troy single-handedly: from leading his army to sacking Troy. And the gods weren't so much concerned about his strength, but his
rage. The whole point of the Illiad is that Achilles is driven by rage and slowly loses his humanity until he ruthlessly kills Hector with the intent of desecrating his deceased body.
I would argue that Achilles's rage works against him in this battle: Punisher will stay cool and execute a plan while Achilles will blindly charge into battle as he did against Hector.
Incendiary weapons were used in Ancient Greece. The soldier would most likely think the gun was some siege weapon. Achilles might raise an eyebrow to the machine in Punisher's hands, questioning Hephaestus on giving a mortal such a weapon, but he isn't scarred for life. Imagine some random child hitting you with something you don't know except that it's loud but harmless. That's Achilles looking at Punisher.
Incendiary weapons are not the same as guns...
And again, the bullets may not cause any grievous injury to Achilles, but that will only make the force of the bullet push Achilles back even more. The Punisher might not be able to do any damage to Achilles (apart from his ankle) with his guns, but he can certainly use them to control the pace and geography of the battle.
How is Punisher going to respond when Achilles runs towards him? How is he getting a shot of his heel then?
How about shooting him with a shotgun that sends Achilles flying back? Again, Achilles isn't immune to the laws of physics, he's just invulnerable. He's not the Juggernaut.
How can Achilles be brutalized if he is impervious to the bullets? You're banking on Punisher getting a clear shot of his heel, which as a marksman isn't a huge leap, but you have him suffering psychological trauma and being physically tortured by things he can't be hurt by in order for Punisher to get the perfect shot in. If his weaponry was shattered into pieces by the bullets, Achilles would go in for the kill. Close range Punisher is as good as dead.
Again, Punisher is far from "as good as dead" in a close range scenario. He's probably killed more people in hand-to-hand combat than Achilles has (he does some of his best work in close range), and he definitely has the knowledge and physical ability to counter a charge even if Achilles managed to pull that off before being knocked on his ass with a shotgun blast or a grenade.
I agree Frank Castle should know Achilles's weak spot. But The Punisher would need a lucky break in order to get to it.
Punisher doesn't need luck. Punisher needs you to start recognizing the shortcomings of Achilles' invulnerability and respect his strengths.
