There are Airport Express ticketing service desks at the airport as well. I can't imagine why you'd not be able to take care of this transaction there. But even if the quoted rule is strictly correct as written, it's not a problem. On the Airport Express, there is no ticketing control at the airport, either as you get on or get off the train. It's all handled at the "other" station. So, if you arrive on a flight and plan to take the AE to Tsing Yi, after exiting Customs you can immediately board the AE without getting a train ticket. Then when you get off at Tsing Yi, you can purchase a ticket before exiting through the turnstiles. This is true whether you buy a regular one-way ticket, the visitor's travel pass you mentioned with cash, or presumably with the advance purchase method you describe.

Also, the ticket you mentioned is available for purchase with cash (and presumably credit cards) at the AE ticket counters at all AE stations (except possibly Expo). There's no need to apply for it in advance unless there's an extra special discount above and beyond the regular offer, which appears to not be the case aside from the 150 Asia Miles bonus.