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Normally I take a USB A to B printer cable and plug the B side into my device and plug the A side into a Windows PC.

When the cable is connected, a virtual COM port appears on the PC because the device with the B side of the cable plugged into it supports the USB communications device class (CDC) and my Windows PC has USB CDC drivers built in.

Will an Airconsole allow remote communication with my device if I plug the A side of the cable into an Airconsole?

I have successfully used a standard USB A to mini-B cable with my airconsole to console into Cisco c3850 switches. I don't know if the Cisco usb console port is CDC though, but I thought I'd let you know.

As Chris mentions, the Airconsole does work with Cisco USB style console ports. These are presented as USB CDC.

Airconsole in general can communicate with CDC based devices but hasn't specifically been tested with non-Cisco devices. Please let us know how you get on!