There are a couple ways to do it. If it was planned to be used often and you wanted a little complexity to the system, run it like a modified encounter. This would allow more use of fate points and the manuvering as per some RTS or Turn-Based strategies in a player vs player type environment. Throwing aspects like "Army out of position" or "Pokemon is poisoned" etc. Then go for the 1-shot "Game Stress" Take out. Even allowing consequences of "On his last pokemon" or "Paralyzed" or something.
If you don't want that to last too long, just go for an opposed roll with the relative modifiers. For a Video Game Scholarship would be the main skill. How well you know the game and its rules, as well as how good you are at handling the controls of the game itself (computer use etc). This base number would then be modified by a whole slew of things. each one adding a +1 to the person who is higher or subtracting 1 from the person who is lower, depending on what range you wanted to keep the final numbers.
Alertness for First Person Shooter,Real Time Strategy, Puzzle,even some turn based games. Seeing what your opponent is doing.
Athletics for Real time strategy games, Fighters(Street Fighter), First person shooters. This would simulate the differences in reaction times etc.
Discipline for your ability to concentrate (or whatever else might be applicable) This would be for almost all types of gaming.
Rapport/Empathy could be used to lay down some trash-talk/make the opponent slip up that way. and could provide various modifiers if the game allowed talking between players.
The TL:DR is Have scholarship be the main skill, but in a video-game world it is modified by so many other factors that even someone who hasn't played in a long long time could still have that "Beginner's luck" or be a "Natural" at the game, just on his own outside skills.
It is worth noting that everything above would be modified a final time by the fudge-dice roll allowing that max of +4/-4 so that there is always a chance that friends beat each other etc etc.
A third option, if the game is of particular importance, is to have a specific skill for whatever game it is. Skill: Video Games (Dinotopia) or what have you. But that affects skill points of the game itself. Maybe have a stunt or something that specifically allows a +2 to scholarship rolls for video game attempts. <--- most if not all kids nowadays would have this. As opposed to adults. along those lines.