Generally speaking, the ogre is going to have a hard time. Frankly, anyone is going to have a hard time with a scroll of antimagic field except a spellcaster that has it on his or her list. The Use Magic Device check is just too colossal for someone to just attach to a random character. Sure, a high-Charisma character that focuses on it can manage it, but scrolls aren’t really intended to be used by fighters.
The best bet is probably to take a level of wizard, or other class that gets antimagic field as a spell. Actually, for a variety of reasons, cleric would be better: that list has antimagic field, being a divine spellcaster lets you use prayer beads, and you can get the luck domain to get a “roll twice, use better” on any d20—that could be your caster level check for this scroll. Plus cleric has slightly better stats than wizard.
Note, however, that for this to work it means Greg must have a divine scroll of antimagic field, and that means it will have to be scribed by a cleric. And since the caster level of the scroll is going to matter a whole lot and we want it as low as possible, we want that cleric to have had the magic or protection domain, so they could scribe the scroll of antimagic field as a 6th-level spell with caster level 11th.
So, once you have at least one level of cleric, you no longer need Use Magic Device at all to activate the scroll. What you need instead is a DC 12 caster level check. Greg’s caster level is only 1st, so this would require a roll of 11—not great, particularly for combat, but we can do things about that.
- The bit of luck ability of the luck domain is worth, approximately, +5. We can determine the precise value of it once we figure out what our final bonus will be.
- A prayer bead of karma (found on a regular strand of prayer beads, 22,900 gp) gives a +4 bonus to caster level. Note that technically this bonus only applies to casting spells, so this requires some fudging.
- The Magical Knack trait gives Greg a +2 bonus to caster level.
- An orange prism ioun stone gives a +1 bonus to caster level. Can be flawed (25,000 gp) without causing any problems.
- With a wayfinder, and similar fudging as with the prayer bead of karma, Greg could get a 1d4−2 bonus—statistically worthwhile (expected value of +0.5), but probably not worth the extra variability, not to mention the gold.
- If you take the arcane domain, the arcane beacon ability gives a +1 bonus to caster level, again only for casting spells. Fudging this to apply to the caster level check to activate a scroll would allow it to apply.
- The Meditation Master feat could allow Greg to meditate for an hour to add a +1 bonus to any d20 check within the next 24 hours.
That is, with fudging, +9 on this check, so Greg fails only on a natural-1. The odds of a natural-1 when rolling twice and taking the better result is 1 in 400. Without fudging, we lose +5 worth of bonus (and save 22,900 gp, on the other hand), so we fail on a roll of 6 or less (30% chance). Rolling twice, that’s a 9% chance of failure (so the bit of luck is worth 21 percentage points, or approximately a +4 bonus).
Note that per the question parameters, you could take three levels of cleric to get another 2 caster level, guaranteeing success (when fudging) and eliminating the need for both the luck domain and the orange prism ioun stone. Personally, I would be willing to spend 25,000 gp to save two levels but that’s up to you. Also improves our non-fudging result so that it fails only on a 4 or less—20% chance on a single roll, 4% chance on a double-roll (at 16 percentage points, the bit of luck is worth approximately +3).
But this is not a winning strategy.
A level, a feat, several magic items, careful planning, and burning a pricey scroll on every one of these ambushes makes this a very difficult, expensive strategy.
What does Greg out of it? A tiny sphere of magic nullification. That he is forced to be sitting in, i.e. any magic items he’s got are gone. Any wizard who lets a no-magic brute get that close doesn’t deserve the title, frankly, and even if it happens, Greg’s got a standard action at best. So he has to somehow get himself next to the wizard—who really ought to be flying anyway—and then do something to the wizard that will prevent the wizard from leaving. Maybe Greg can do that—but then Greg can probably do it better with magic in play.
Basically, antimagic field is a trap. It hurts the person casting it far, far more than it hurts anyone else. The only people who should ever use it are arcane archers and the like, who get the ability to cast it over there and thus not screw themselves over in the process.