5000 BTU is about 1500 Watt Hours which would take about 12.5 A at running mode. Start of the air-conditioning unit could peak at 50-100% more.
Honda 1000 maximum wattage is 1000 W with 900W (7.5 A) running mode. This is about 66% of your need at the running mode, sorry.
I have Honda 1000 which was insufficient to run 800 W microwave
due to start-up power requirement.