He would probably never get it. Ignoring temperatures etc. the can would accelerate towards the center of the Earth, gaining speed until it hits terminal velocity. Once it passes the center it would start being pulled back towards the center. Think of it as a pendulum, starting at one height but never quite making it back to that same height as is swings back and forth until it eventually comes to rest at the center of the Earth.
If the air were evacuated, there would be no "terminal velocity" as long as gravity is pulling it towards the center of the earth, unless there is another force acting on it such as friction with the walls of the hole. It would continuously accellerate (or decelerate, which is the same thing) any time it was not exactly at the gravitational center of mass of the earth. On the way "down" it would accelerate continuously until it passes the center point, at which it would begin to decelerate (accelerate in the opposite direction) until it came to a halt (in a perfect system it would rise to the exact height as it was dropped from), at which point it would start back down the other way.
What we're really quibbling about are what factors prevent it from being a perfect system. If it were a perfect hole evacuated to a perfect vacuum and the can fell with no friction from the walls or other factors causing it to lose energy, it would cycle back and forth forever. But no system is perfect, and even a miniscule amount of wobble in the earth's rotation and the precession of the earth as it zings around the sun would cause the can to press up against the wall of the hole just a tiny bit. That factor alone would be enough to gradually transfer the can's energy into heat, even if the rest of the system were "perfect" with no friction or other sources of energy loss.
Oh yea, don't forget those other sources of loss. If the can is metal and passes through rocks with different magnetic or electrical properties, magnetic flux forces will be generated which will again transfer energy to heat.
So, we're quibbling about sources of energy loss, because although in a perfect system it would oscillate forever, the system simply cannot be perfect if for no other reason than the earth is tipped a bit on its axis as it rotates around the sun, and that would cause the wall to have to exert some force on the side of the can to keep it going up and down the hole. And that's just one source of energy transfer/loss.
Then again, if you build the can right, some of the energy lost to heat could be used for propulsion. Having one end of the can a better radiant surface, for example, would create a miniscule amount of thrust as the heat created by the friction was radiated from the can. For that matter, you could use energy transfer from the hole to the can to actually ADD energy to the system, so the can would actually oscillate farther and farther each time through. If the walls of the hole transferred heat energy to the can, and the can was designed so that the radiated energy was always directed in the direction that the can was travelling, then the can would accelerate faster than the acceleration caused by gravity. That additional acceleration could be calibrated to exactly counteract parasitic losses due to can-hole friction, earth wobble, losses due to magnetic flux, etc.
