Like the characters, most items and weapons of Zelda Cube remain a mystery. Only several are confirmed, and they are the obvious ones.
Link will have a sword, as usual. Not sure about what sword was shown in the screenshots, could be the Master Sword, we'll have to wait and see. Overall, though, I think it's obvious that Link will have a sword throughout the game. It's been his trusty weapon since the original Zelda began.
And also, the Hero's Shield/Hylian Shield will also make an appearance. You can also count on the Kokiri Tunic and Kokiri Boots, the standard outfit for Link.
Ganondorf had a huge sword in many of the original Zelda Cube screenshots, but any chances of obtaining this colossal weapon have been ruined with the switch to cell-shading.
These are the only items/weapons that are confirmed to be in the game. However there are others that we can guess. There are many regulars that appear in Zelda games quite often. These include bombs, the bow and arrow, Deku nuts/sticks, the boomerang, the hookshot, etc. You can probably expect some Ocarina of Time and Majora's Mask items/weapons to carry over onto Zelda Cube. Probably some new stuff that we haven't seen before also.
As more items and weapons are confirmed, this page shall be updated.