Is time going to slow down for us when the expansion of our Universe accelerates?

According to the theory of relativity this is already happening. The reality of it is that there is no "twin" in any position to see the difference. If time relative to earth changed it's rate of page, even if by a large amount, we would not be able to see it because we would change with it. It's all in the difference between two points of view. With all humans on or near earth, the chance any sort of change would be observable is very, very small.