Will a 1000 watt generator operate a sump pump?

That depends on the power requirement of the sump pump. A 1000 watt generator (if this is running watts) will produce 1000 watts continuous. Through some simple math, this is equivalent to 8.33 amps at 120 volts. Current (in Amps)=Power (in Watts)divided by Voltage (in Volts). On your sump pump, there is something called a nameplate which lists model number, serial number, manufacturer, and power requirements. The power can either be listed in watts directly, or in amps (at 120v). If it lists watts directly, this number is either higher or lower than your 1000w generator. If it lists amps, your generator will supply 8.33 amps continuous, as figured above. Likewise, you can find out if your generator can power any given load by using this method. Just divide the listed wattage by 120 to get amps. Also, motors do pull higher current when they start, so it is usually recommended to size the generator larger than you would otherwise when you are running a motor, such as your pump. If the sump pump is right up there at 8 amps, it would be pushing the limit to expect it to run the pump. Some smaller generators too are so-called "inverter" units, and many of these are not recommended for motor starting duty. Check the generator's manual to be sure.