Box texture mapping will map that correctly without any need to add any background or adjust UV tiling settings. There’s a problem, however, with your PNG texture map which is that it has extra image information outside the image itself, i.e. transparent pixels at the edges, which make the image shrink and have a black border around it when applied as a texture in Element. So you need to remove the extra transparent edges, in Photoshop, by trimming off transparent pixels, before bringing it into AE.
Kalleheikki Kannisto
Senior Graphic Designer