How much power will your bug zapper use?

Not much. It depends on the unit and the design. Some only use the wattage of the bulb (plus any ballast losses), and only use more in spikes when it is actively killing bugs. Others use a step-up transformer to power the killing grid, and the transformer adds to the voltage losses. So expect anything from 30 to 200 watts plus momentary spikes. You would have to consult the label and specifications for your exact unit for more details.