Knowledge of electrical potential and energy is important to systems where electrostatic forces play a role. We calculate exactly the electrostatic potential of a uniformly charged cylindrical shell at an arbitrary point in space as well as its electrical energy. The expressions derived are applicable to cylindrical models containing charged particles and certain biological systems with cylindrical symmetry. An explicit analytical formula is provided for the total energy in terms of a class of special functions known as generalized Hypergeometric functions. The expressions when written in one-dimensional integral form are very suitable for numerical calculations.