Why would anyone ever, ever say that? Please, nobody ever say that.
One-upmanship is the act of surpassing another person. In this case, one female character is one-upping her friend's claim of being taken on a mountain hike with a claim that she was proposed to.
Mario is the major figure in the Super Mario series. In the games, completing specific conditions causes a "1-up" (but the marks are chevrons («»), used in some languages like French or Russian instead of quotation marks) to appear on screen, referring to an additional life. The comic relies on the homonym of the action of one-upmanship and the event of one-ups in Mario.
In the title text, Randall says that this is a pretty bad joke (and also somewhat tasteless) and he doesn't wish to hear it.