Coach from sofia to plovdiv

I am not sure about from the airport, but from the main bus terminal, (which is across from the main train station) there are buses that leave every hour from sofia to plovdiv... more frequently during the busy hours. A taxi to the bus terminal should cost you around 10-15 bgn.

There might be a place to catch them on their way out of sofia (as the airport is on that side) but the sofia-plovdiv line has a tendency to be rather full, so even if it does stop on the outskirts of the city, it is not guaranteed they would have a free space, so it is better to catch it at the main terminal.