The reason for the clock to be out that arrived on the rocket could be due to effects of previous DHARMA Initiative experiments with time travel in the Orchid station, perhaps the whole islands' time is 31 minuites behind and this is why the rocket took some time to arrive.
Ever since The Swan imploded, the time difference between the Island and the outside world has been increasing.
How do we know the time difference is 31 minutes and 18 seconds and not xx days/weeks/months/years plus 31 minutes and 18 seconds? Did anyone bother to check the date?
That info wasn't provided in the episode. All we have to go on were the brief glances at the two time keeping devices, which didn't appear to display the date. I would also argue that a half-hour discrepancy, as opposed to multiple days, could be easily overlooked by a travelers, especially when you land in a dangerous and unfamiliar locale such as The Island.
Daniel is talking to Regina on the Satellite phone and hears that from her perspective the payload has landed, but doesn't see it on his end until a short time later. Based on the conversation and Daniel still sitting and waiting for it, 31 minutes seems more reasonable than several days/months/years plus 31 minutes since Daniel didn't sit and wait for days/months/years.
After the doctor washed up on the beach, it confirms that the time discrepency would not be of years. If his body had been in the ocean for a year, there would be signs of decay.
If the "Island time" is 31 min off, then Daniel's wristwatch should be off 31 min as well, assuming the beacon and the helicopter followed about the same path onto the island.
I imagine Daniel's watch was broken by the electro-magnetic field surrounding the island when he jumped out into the storm. (see Jin's watch) The beacon had a metallic casing that would prevent any such damage.
The fact that there is a delay in time between the outside world and the Island, is likely how the writers decided to write it Walt's faster aging. Since the actor in real life got older faster than the show could keep up with.
See the first diagram to the left for a possible explanation of the results of Daniel's experiment.
The different perception of time isn't necessarily far forward or far backward, it may be parallel timelines that move at the same rate, only the island's may be slightly behind the outside world's, and the island simply has to "catch up" to the rest of the world before things can appear in the "island's world". This would cause some kind of limbo between the time barrier of the island and the rest of the world. Gas would still be used and motion would still be presumed, but these things would not ACTUALLY be happening. This further explains why Oceanic 815 was not found, since the plane was only about 1 1/2 hours off course or 750 miles. The reverse, going from the island to the outside world, would cause someone to have to either jump backward (which would explain Desmond's falling "too far back") or possibly jump "sideways".
See the second diagram to the left for a possible explanation of the the phenomenon mentioned above.
The Hatch implosion caused the distortion of time between the island and the world. When Danielle Rousseau is pregnant with Alexandra, she claims the year to be 1988. In 2004, we see a 16 year old Alexandra. No continuous time shifting or difference between the Island and the outside world could have occurred. 16 years on the Island (experienced by Alexandra) and 16 years off the Island (2004 minus 1988).