You do not want an H bridge to control the servo. An H bridge will work for a DC motor where you are directly controlling the power to the motor, but not for a servo like this where the servo assembly has it's own internal H bridge and motor control.
The servo is controlled via a pulse modulated signal that tells the servo the speed and direction. Low pulse durations turn the servo one way, high pulse durations will turn it the other way. Read the specs for the exact pulse durations. Note that the key to this is that it is not a "duty cycle" but a "pulse duration" system - the time between pulses is not very critical.
Power for the servo can be separate from the control input as long as they share a ground - so your Arduino can supply 5 volts while the actual power is 6 volts