The way I see it,
If the rod was to be tapped and a compression wave traveled down the rod it would travel the same speed that sound travels in that medium, a lot slower than c.
If the rod was pushed by hand then it would accelerate according to the force applied by hand,say at a rate of .5m/s^2.
So if the person at the other end required the rod to be pushed into his hand .001 m (so that he could feel it) and the rod accelerated at .5 m/s^2 then the time taken for the 'information' to be sent, t=sqrt((2*.001)/.5)=>.063s.
The only way the 'information' could be sent faster that c would be if the rod was accelerated faster than c => the hand must have and infinite energy source to do so.
The only way the 'information' could be sent instantaneously would be if the rod had to travel only 0m for the pulses to be noticed, which is impossible.
PS. Sorry if i am alittle confusing, kind of hard to explain what i am thiking