Final Round...!

I'd play the World's Hardest Game. I ain't going down without a fight.

 

Golden Sun, without question.
If I want the world to end on a positive note, I will have my last memories of such be pleasant. I'll play my favorite game as everything is incinerated.

...I might not be able to finish the game before the world concludes (it involves about 20 hours of play from start to end, after all), but I could try to get as far as I could.

 

I think I'd play The World Ends With You. I've never played it, but I've heard it's a great game. And it'd be a bit of situational comedy before I die. xD

 

Pokémon Blue.

The series has been there for me since its inception when I was only a lad. What better way to die than have that last moment be you and your childhood; one more time together. Still glad I have my GBC in case an event like that occurs.

If not Pokémon, I'd choose Majora's Mask. Although I prefer OoT (as it's also my favorite game of all time), MM is obviously a lot more fitting in the apocalypse scenario. I'd try my best to match both Link and I's deaths together in sync.

 

I recommend it, largely because of the plot. It's an RPG, so the post-game and grinding to get all of the goodies is a bit of a pain. However, that is not really what defines the experience of that game. All you really need to do to enjoy it to its fullest is to finish the story mode, then collect all the secret reports to get a complete picture of the plot.

 

Minecraft. I have many awesome memories with friends on that game and I'm sure creating a world with them once more before it's all over would not be a waste of my time. We'd go through everything, from the first night to The End. We'd have so much fun, stupid moments that we'd be focusing more on the game than what's happening in reality.

 

Can't I just take all of them to death with me?
I'd say Eternal Sonata.

Spoiler

Spoilers
The game literally ends like the same fate you have, the end of the world. And Chopin's death.
The beginning of the end.