What is this manga about a guy who completed a video and he gets a mysterious game?
Gamerz heaven

From wikipedia:


  Soon after starting the game, Kaito is attacked by the class president, Ogura. Ogura falls to Kaito and vanishes from the Second Zone. Now no one believes that Ogura ever existed except for Kaito's best friend Kawashima, who loves video games just as much as Kaito. Later Kaito eventually convinces other friends Rio, and Ren, who didn't believe him until a "meteor" hits the center of Tokyo, which was actually the work of the first area boss of Gamerz Heaven, Rush.


Regarding the mysterious girl:


  Upon reaching this strange new world, Kaito discovers a boy known as the "navigator", aptly dubbed "Nata". In Gamerz Heaven the real world is referred to as the "Second Zone". All enemies in Gamerz Heaven are after Nata and Kaito can only beat the game by saving him.