When Boson gas approach lambda point (no matter from superfluid side or liquid side),the heat capacity tend to diverge, doesn't this mean that we can never drive the gas through lambda point?(cause we need infinite amount of energy to do so)
The amount of heat added to the system is the integral of the specific heat wrt temperature:
$$ Q = \int C(T)dT $$
So in the link you give it's just the area under this graph:
Although it's true that the specific heat tends to infinity at the lambda point it does so sufficiently suddenly that the area under the graph remains finite. That means the amount of heat needed to move the system through the lambda point is finite.