Heavy metal is godly may be a fairly accurate description of Eddie's power.
Lets think about this, P walks in carrying a heavy bag. If it was me I would probably carry some sort of antitank sniper rifle(4 rounds), a machine-pistol(60 rounds), a pair of heavy pistols(20 rounds), maybe 5 stun grenades, 2 incendiary grenades, and a remote-detonated high explosive. The bag could contain spare ammunition and traps, probably land mines. There would also be various detection equipment and supplies. I would look for a sniper position, something with good visibility but not obvious. I would mine the area, hide the bag, and watch for the opponent. If the rifle doesn't work I would leave it behind. When they get close I would toss the bomb somewhere I could remember and switch to the stun grenades and machine-pistol. If the pistol is ineffective I switch to the heavy pistol, if it looks disabled in any way I shoot it a few times with a heavy pistol and lob an incendiary, then recover the bomb for good-measure. If nothing seems to work I would try to lure it back to the bomb...
Eddie will probably fly low, searching for an opponent, at which point he would manufacture lightning and then close in for melee after having some fun with that.
Flight will render mines ineffective, a point-defence device would be beyond the scope of the tournament, the rifle may be a deciding factor, depending upon the ability to hit a moving target, grenades would be difficult to use against an airborne target. I would say that the first attack would be some sort of rifle against Eddie, then lightning against the Punisher, then heavy pistols against Eddie, then axe against the Punisher, then fire and/or high-explosives against Eddie, then more axe work and pistols.
I am not really familiar with these characters, so I can't give accurate specifics...