Indeed Lugia is not water type, besides being a Water Guardian.
Kyogre does have ways to destroy Lugia, since Lugia is Psychic/Flying, Kyogre can use his Electric, Ice and Rock moves, and recover is HP, using Rest.
Lugia only can use his Electric moves to attack Kyogre and use Rest and Giga Drain to regain HP.
Kyogre appears to be the winner
A good tactic for Lugia is to use Thunderwave, to cause paralsysis, and then a full charge with his Electric attacks or GigaDrain.
Comparig Base Stats:
Lugia- HP, Def, Sp. Def, Spd.- 4
Kyogre- Atk, Sp Atk.- 2
Lugia has better HP, and his more defensive and faster then Kyogre.
Kyogre is a better attacker.
In a combat Lugia even it a lower attack rate comparing with Kyogre, Lugia doesn't lose much HP, because his better Def and Sp. Def, and has more ways to regain HP, Lugia attacks are more effective because Kyogre low Def and Sp Def, and Kyogre only have Rest to regain health, so the winner of the seas is Lugia.