My Time at Sandrock

The museum reward for donating 80 items is 500 game center coins, this also unlocks the achievement.

There are lots of things that can be donated, as well as the relics collected by mining. If you hover over/check item descriptions and something has "In museum small exhibit" (or medium/large) that means it can be donated. There are clothes such as the desert defender set that count, tools also count, most weapons count. Then there are some machines and items that can be built on the assembly station that can be donated too.

Once you get a certain way through the story, the game centre will get a punching game (Punch a tron).

Keep playing that, aiming for the middle and fill power. I'd been playing loads and it took me two in game days of non-stop playing to unlock achievement.