Since a Rubik's cube is a figure with borders, you can't have a Rubik's cube with infinite sides. Even theories about the universe not being infinite usualy say that it has no "borders".

A way to trick the problem to get an answer would be to assume that you are yourself infinitely smaller than the Rubik's cube. So the object would be infinitely bigger than you (as it was stated in your question) BUT it will still be part of our world at its normal size. So the answer to your question is: no you can't solve the Rubik's cube but someone else can.