If pure-blooded Garleans can't use magic...

[Enkidoh]

I just took it that his armour was magitek enhanced (I mean, he has a magitek cannon mounted on his wrists!), and given the fact that part of the Praetorium was smothered in aether (Cid clearly stated this, and it's the reason why the linkshell transmission between him and the player was lost), his armour is probably designed for this very purpose - to utilize whatever aether (or cereleum, the processed form of aether) it can pull from the surrounding environment, something not too dissimilar to how magic in Eorzea actually works (and this fits the in-universe information about the Garleans developing magitek to actually compensate for their lack of magical aptitude - it makes sense that magitek would actually function like magic).

Hence it's likely his armour simply drew the aether from the surrounding air and empowered him. Not that it helped him really - he still folded faster than Superman on laundry day.