The answer is no, in fact.
General and special relativity are complex beasts.
General relativity basically states that the deeper you are into a gravity well, the slower time appears to go from the point of view of an outside observer. Black holes, the deepest gravity wells of all, have what is called an event horizon. As you approach it, time appears to stop for someone observing you - that is, the last few feet of travel will apparently take a billion years, but for you ity will all be over in an instant!
Special relativity is the opposite. The faster you are moving the faster time appears to go from the point of view of the one travelling. This effect only comes into play at c-fractional velocities. Superman not withstanding, it has nothing to do with the direction of travel.
I actually did the maths for a sub-light-speed voyage to Alpha-Centauri and back once, for a game I ran called "To The Stars, The Slow Way".
A space craft accelerating at 10m/s2 will reach the speed of light in 29979246 seconds which is 347 days. Of course E=Mc2, so it is actually impossible to reach the speed of light, and as you approach it the energy needed to accelerate further increases exponentially. But for this purpose, let's assume that things are linear to make the maths easy.
So, it takes light 4 years to travel from Earth to Alpha Centauri. We can assume that, travelling from a standing start (relative to the Earth) we have travelled pretty much half a light year in that year (near enough) that it has taken to get as close to the speed of light as possible. When we get to within half a light year of our destination, we will need to turn around and accelerate backwards in order to slow down to a stand-still relative to our destination. This will take exactly the same amount of time as the acceleration at 10m/s2. So it takes a year to do the first half light year, a year to do the final half light year (give or take), and three years (near enough) to do the three light years in the middle. So from an observer's perspective it takes five years for us to reach our destination.
However, the crew of the space ship will experience time dilation, and for them, the middle three years will pass in a fraction of a second. The entire journey for the crew will take about a year due to the increasing effect of time dilation as they get faster.
It is interesting to note that no matter how long the journey actually takes, for the crew it always takes about a year, because the time spent at near-c will always pass in a fraction of a second no matter how long the ship is at that speed for!
Now, I have a question....
If I am travelling at just under the speed of light and fire a laser ahead of me, does it go anywhere?
Equally, what happens to a laser I fire behind me?
Work that one out if you will?