I know airplanes have an aluminium outer body that lets lightning just slide through the outer surface and not causing much damage. But what will happen in the case of a zepplin?
Firstly, Zeppelins usually use a conducting shell. Secondly, inside the hydrogen-filled shell there exists "no" oxygen. Hence, it is not possible to ignite a fire inside the shell. Nevertheless, Zeppelins fires have been observed during lightning strikes. However, this was during the landing procedure, when hydrogen is vented to increase the ballast. The vented hydrogen mixes with the oxygen in the atmosphere, creating a combustible mixture.
Sidemark: One hypothesis for the famous German Zeppelin "Hindenburg" disaster is a lightning strike during the landing process.