If the airline changes my customer's flight time after I have issued the ticket, will I find out or will they only know at the airport?

Yes, in most cases you will know before your customer does. When an airline moves a flight, it files the change against the booking, and your supplier passes that back through the flight booking API, so the new timing shows against the booking in your system instead of only turning up on the day of travel.

How fast you see it depends on the supplier and on the way your connection is set up. Some suppliers push the change to you as soon as the airline files it. Others expect your system to look at the booking again and pick up the new times, which means you find out on your next check rather than the same minute. Low cost airlines often work outside that flow and may write to the passenger directly, so your customer can hear about it around the same time you do.

What you are allowed to do about it is the airline's decision, not something your portal controls. A small shift in the departure time usually just needs a polite call to your customer. A bigger move, a different day or a flight that is dropped altogether, normally lets you put them on another flight or ask for a refund without a penalty, but each airline sets its own line for what counts as big enough, and your supplier follows that line.

Ask your supplier whether schedule change alerts come with your plan and where they arrive, and keep the habit of checking bookings again in the days before travel using contact details you know are still current.