According to posts made here in the past, that Honda 2000 will work fine. However, always keep in mind that a 5000 BTU A/C will operate typically at around 700 watts just running the fan, but will surge up to 1500 watts or more when the compressor kicks in. It's still well within that 2000 watt operating range, but just be frugal when running the A/C and anything else.
The new inverter style generators like the Honda work well with consumer electronics like computers, TV, and such. A better choice than cheaper non-inverting gensets. But if you run your electronics with a voltage conditioner device like a UPS or other conditioners that are available, you can use a cheaper, non-inverter type generator safely. A UPS, (uninterrupeable power supply) isolates your electronics from the line voltage by actually running your gear from a battery
& inverter curcuit. Voltage spikes, brown-outs, and dirty AC has no effect since it's not felt on the curcuit your gear is on.
Thats not the case with a "Surge Protector" type device that only protects by popping a breaker if there's a voltage or current surge, and they offer no protection for a "under-volt" condition, like when a generator loads down or drops off.
If you can afford them, Hondas & Yamaha are the way to go, but with a additional purchase of a good UPS or a dedicated Voltage Conditioner, a standard generator will do the job just as well.