Can batteries conduct from their sides?
With most single-cell batteries (e.g. AA), the case is connected to its negative pole.

A battery by itself does not provide enough voltage to feel by hand. However, the flash circuitry charges one end of the flash tube to a high voltage (between 250 and 5000 Volts, depending on the tube). This energy is discharged as soon as the trigger circuit is closed, which your hand did.

Make sure electrical systems are disconnected from power, and make sure high-voltage systems (such as flashes) are completely discharged before disassembling them!