This is more of a philosophical question rather than a physics one, but I'll give it a shot :) First, a word on Time:
Time and space were created in the Big Bang - therefore time has indeed been around forever. There was no 'time' prior to the Big Bang, therefore there was no 'before' the Big Bang. It's a bit of a circular argument unfortunately - time has been around 'forever' because 'forever' requires time to exist since 'forever' is a measurement in time.
This is relevant to your question. If we were to increase the expansion rate of the universe, and thereby alter the rate of the passage of time, how would we know we had done it? We are within the universe and subject to the passage of time, so any change in the rate of the passage of time will affect us too. Only an observer looking in to our universe from outside would be able to see any differece.
Here's an example: One second is the time it takes light in a vacuum to travel 3x10^8m. Before we expand the universe, we mark out 3x10^8m and measure how long it takes light to travel this distance, giving us one second. We repeat the experiment after beginning the expansion. What do we find? It has taken light one second to travel the same distance. The problem is that we (and our clock) are within time. Our outside observer will notice that the 1st second and 2nd second are different lengths, because he has not been affected by the change in rate of time. Inside our universe, time occurs at one second per second. Everything in the universe ages at the rate of one second per second.
For us to travel to the future by changing the rate of expansion of the universe, we would need to pop out of our universe while the change was happening - we would need to become the outside observer so that we could measure the difference in the rate of time. Then we could pop back in to the universe at a later time. To put it another way, the universe and the traveller will have aged at different rates.
Another way of thinking about it: consider a movie on cinefilm. The movie shows the light-measurement experiment described above, and it shows it twice. We play the movie so the first experiment is played normally, then we speed up the movie before the second experiment. The movie represents the universe, and we are the outside observer. To us, the movie looks speeded up. On the cinefilm though, each frame is 1/24th of a second of the movie.
Of course, combining my answer here with Hawking et al's hypothesis that time will reverse when the universe starts shrinking means that the universe will end when the shrinking begins, at least as far as the occupants of the universe are concerned. An outside observer will see the movie stop and then be played backwards. Time has been played out, nothing more can occur in the universe. No new memories can be formed, nothing more can happen, not even an awareness that time is running backwards because only an outside observer can observe that.
(Note that since the original question asked about changing the expansion rate of the universe, I have not addressed relativistic time dilation since this is a local phenomenon rather than universal. It would of course be much easier to travel to the future by local time dilation than by altering the universe's expansion).
