The wooden bat is a decent weapon, capable of smashing open the skull of Them, despite it being made out of wood. The wooden bat is slightly weaker than its counter part, the metal bat. But unlike a metal bat it is more durable allowing the user to use it longer since with repeated use a metal bat will eventually dent and bend due to excessive use.
Also by putting nails in it you can deal additional damage since the nails will penetrate the skull and pierce the brain of a zombie.
The wooden bat is Shimada's primary weapon, and he uses it to save help the boy who led the infected into the mall. However, Shimada is unable to defeat the infected, and he is bitten and is killed by Asami.