1) The only thing unique to the Priest/Priestess that I can find that has a bonus vs non-living creatures is the Staff of Asyra. Besides that, there are a few Holy spells that also have such a bonus, but any mage can use those including future Holy mages, it just happens that the Priest/Priestess is currently the best at utilizing them. Those few holy spells are Samandriel, Pillar of Light, and Blinding Flash. That is all.
Pillar of Light and Blinding Flash are not a huge concern. The bonus vs nonliving means Pillar of Light will be doing equal initial damage to your undead as a Flameblast would (and then you compare the benefits of Daze vs Burn). Meanwhile, Blinding Flash is still doing less damage than a Ring of Fire. I don't believe that these provide the Priest/Priestess with a significant advantage.
So, you're left with the problems of Samandriel and Staff of Asyra. These probably are more powerful vs a Necromancer than similar cost/level spells are. However, they are only 2 spells and can not be duplicated within the arena (as one is legendary and the other is equipment). I might make them priorities, but I would deal with them the same way as any big beefy creature or annoying equipment.
2) Flyers can present a problem for any mage not prepared for them. Some mages may have more ways to combat them than others, but they all have ways to do so. I would recommend reading this strategy guide on how to combat flyers
http://magewars.com/jsite/strategy-guides/item/84-strategy-guide-example and then adjust your spellbook/strategy.
3) Then don't solo with him. Play to his strengths.
4) I will agree that zombies are not great for an aggressive rushing strategy and your opponent will have time to build up if you use them. If you want to use zombies, I recommend that you adopt a more long term plan rather than rushing. I posted a suggested build that would love to face an opponent that gave it time to build up, in which by turn 3 I was producing 17 mana per turn and 2 creatures + a quick spell per turn, without having cast any mana crystals or harmonizes. Give me some time to get those out as well, and I believe I could out produce the Priest/Priestess.
On the other hand, if you don't want to compete with the Priest/Priestess in a long, slow game of out developing one another, you could try using different creatures. Skeletons don't have the same inherent weaknesses, and don't forget about the other creatures at your disposal. Adramalech doesn't take up many more spellpoints for a Necromancer than he does for a Warlock.