In Spain we have good communication via trains, especially with the high-speed ones (Ave and Alvia). Ave can be a bit expensive (faster), and Alvia less expensive ( It is not as fast as Ave is, but still you can make Madrid-Leon in 3 hours). Anyway there is no Ave between Madrid - Leon (yet), you may find special offers on trains if you book in advance.
And then for sure, you will find a bus that will fit with your schedule to do Madrid-Leon.
I am Spanish but living abroad, so I used to do Madrid - Leon 4-5 times per year, so basically for the bus there is more frequency and you can take it directly from Barajas (T4), train less frequency but faster... But if you go with some bags, it is handy just take the bus directly in the Airport.
Then in Leon, both bus terminal and train station are pretty close together, and 10 minutes walking you will reach city centre.