Hi Bethc!
There aren't direct trains from Madrid to Astorga on Saturdays. Your arrival time is late for options changing trains along the route (you would need to take a train at 14:40 out of Chamartín train station to make all the route by train on your arrival date).
Bus, as you said, is an option. In fact, it's the easiest option because you can take it at the airport itself. Take a look at the web of Alsa (
www.alsa.es/en) for info about schedules. If the schedule of buses out of the airport doesn't suit you, there are more frequencies out of downtown Madrid (Estación Sur de Autobuses) but you would need from the airport to the downtown by yourself (and don't forget to take into consideration the time it takes to go from the airport to Estación Sur de Autobuses when planning). Tip: the metro and train station for Estación Sur de Autobuses is Méndez Álvaro.