After a bit of thought, I realized I could probably make my reply more concise. So, I did some editing.
If you want to see the original, unabridged version of that post, which I feel goes into more detail, I have saved that & made a Pastebin of it for your viewing:
https://pastebin.com/qHpZqJ7r
ANYWAY....
Zapdos has: Better AP, SS, Durability, likely better flight abilities, & possibly equal or better stamina &/or intelligence.
"Special" (Non-physical) durability buffing via Light Screen (temporarily halve Special attack damage Zapdos receives.) & Charge (Buff Sp. Def)....
If it has Lightning Rod, Drawing in & absorption of Lightning via Lightning Rod, which also buffs its Special Attack, which in this case, helps its Lightning attacks & Ancient Power, Ancient Power being the more relevant one. The Lightning Absorption also means using Lightning Body while close to Lightning Rod Zapdos carries a risk, as LR lets Zapdos draw in and absorb electricity, & get a power-up from it.
A likely Stamina advantage if it has Pressure, thanks to Pressure meaning Laxus will expend more stamina fighting Zapdos than he would otherwise, & thus, probably tire out first.
10% chance of buffing all its stats via Ancient Power.
Detect to allow it to evade attacks.
Roost to heal itself, as well as temporarily make itself resistant to Lightning, as opposed to Neutral, as is usually the case with Zapdos's Electric/Flying typing.
The above defensive options mean that Laxus using the Legendary Spell Fairy Law could be evaded, tanked thanks to durability buffs from Charge, Light Screen or Ancient Power's random chance effect, & if Zapdos has Pressure, casting Fairy Law will be even more taxing than usual.
A possible advantage through Zapdos being a more adept & flexible flier, & thus, better able to make use of Tenrou Island's terrain.
Zapdos may have lightning immunity/absorption regardless of which ability it has.
http://bulbapedia.bulbagarden.net/wiki/Zapdos_(Pokémon)#Pok.C3.A9dex_entries_2 To quote one of its Pokedex entries:
Zapdos is a legendary bird Pokémon that has the ability to control electricity. It usually lives in thunderclouds. The Pokémon gains power if it is stricken by lightning bolts.
Laxus has: More adept at close combat, as he has more limbs he's used to attacking with. Thus, he could overcome Zapdos's superior AP, SS & Dura through landing more attacks in close combat.
Thought Projection could possibly be used for decoys, but the use may be limited against Zapdos & its "high" intelligence. Lightning Immunity means Zapdos will have to mainly use Ancient Power for its ranged offense.
Lightning Body allows him to use his magic to turn himself into Lightning, & thus fly, & have a level of intangibility. His own page, however, suggests that it can only be maintained while he is in motion.
Possible advantage from fighting on Tenrou Island. Quoting its page on the Fairy Tail Wikia:
"The Great Tenrou Tree sinking its roots in the center of the island bestows divine protection upon those bearing the Fairy Tail crest, making it so that they cannot die while on the island, while at the same time enhancing their Magic Power. Such effects would be nullified in the unlikely case that the tree was destroyed."
Could potentially use his Lightning Absorption to eat electricity produced on contact, if Zapdos has Static. This would allow him to gain healing &/or stat boosts.
The advantages I listed above means it can buff its special durability enough to not mind electric attacks if it isn't immune already, & potentially tank Fairy Law, not to mention, Pressure making Laxus have to use more stamina than usual makes Fairy Law even harder on the caster.
Add to this healing & temporary Lightning Resistance (If it isn't already immune.) via Roost, evasion via Detect, & Ancient Power's 10% chance of buffing all of Zapdos's stats, which Zapdos will likely be using more once it realizes Electricity isn't hurting Laxus.
Zapdos likely having the better flight, & thus, better mobility as well, means Laxus will have trouble keeping things close combat, & even if he does, Zapdos can still fight back with Ancient Power, Drill Peck, & evade with Detect.
If Zapdos has Static, its chances are a bit worse, as Static doesn't really help at all -In Pokemon, electric-types can't be paralyzed; Laxus naturally uses electricity, so it's very unlikely it'll affect him- & Laxus's Electricity Absorption may allow Laxus to BENEFIT from it.
But Lightning Rod or Zapdos being otherwise immune to electricity means Laxus has to rely on Fairy Law for ranged offense, & Pressure means he loses stamina faster, which is bad ranged or close up.
Both can fly, but Zapdos is likely the better flier, & thus more mobile. It's also better outfitted to outlast Laxus in its abilities, it can be close, equal, or better than him in several stats, & to me Laxus seems like he needs to take advantage of his advantages in close combat against Zapdos, which just doesn't seem easy here, all things considered.
Whereas Zapdos can still fight at a range or up close, & has help from Detect, buffs from multiple moves, possibly Lightning Rod, & possibly Ancient Power buffs, plus healing with Roost if it gets an opportunity.
Even if Laxus has a buff to his magic power by being on Tenrou Island, Zapdos may still be able to overcome it with Charge, Light Screen & possibly Ancient Power's randomly occuring effect allowing it to tank Laxus's non-physical attacks better.
As for preventing death, its uncertain if this offers Laxus extra durability, or just prevents him dying. It's uncertain how that would matter if it doesn't affect durability, as an opponent's Death is either a win condition -in which case, doesn't that mean Laxus can't lose here?- or irrelevant.
'
And all that assumes that Tenrou Island DOES benefit Laxus, as the divine protection is specifically bestowed upon "'those bearing the Fairy Tail crest", which I'm uncertain if he has; I'm not sure if he HAS the "Fairy Tail crest" on his body.
Overall, I believe Zapdos wins this, with high difficulty.