When we say that the universe is flat, we don’t mean the physical geometry, but rather spacetime itself. The easiest way to think of this is that two parallel beams of light shone into space will continue to travel parallel forever if the universe is flat. Two parallel lines on the surface of the Earth which is not flat would eventually meet and cross over.
Being flat would mean that everything would keep moving apart slower and slower. But as the universe gets older light from distant objects has had more time to reach us, so we can see further. I am already not sure we can see anything outside this galaxy with the naked eye, but things will definitely get fainter.