Urusei Yatsura ~Dear My Friends~

Urusei Yatsura: Dear My Friends is a point-and-click adventure game based on the anime and manga of the same name. It is the story of Ataru, a high school lecher who loves nothing more than to try his luck with every pretty girl he meets. Although he is generally good-hearted, he does a good job of hiding that fact.

One day, Earth is beset by alien invaders who will conquer the planet, unless a chosen representative is able to beat an alien representative in their national game: Tag. Incomprehensibly, Ataru is chosen for the task. His opponent is Lum, gorgeous daughter of the invaders' leader. Shinobu, Ataru's long-suffering girlfriend, tells him that she could never marry a loser, and that he had better win the game of Tag if he wants to marry her. Ataru is filled with new hope, and against all odds, he manages to tag the flying Lum.

[From Mobygames]