If there's a bus at Barajas that goes to Leon that makes sens with your arrival time, why not look into that. Will save you the comute downtown.
If you are too tired or it's too dark when you arrive in Leon, and you want to stay in Virgen del Camino, you can take a city bus on the large round about by one of the banks.
Not sure what albergue you are considering in Virgen del Camino: Don Antonio will ne closed, according to Eroski. The one in Valverde de la Virgen, further along, will be open and you may make a reservation, again according to Ersoki.
Further along, I would avoid the Ave Fenix albergue in Villafrance del Bierzo, if not just for the accomodation then for the owners attitude.
Ah, my story about Hospital ... My feet had been killing since Leon. Flat feet I didn't jnow I had. So in the morning I popped in the first bar I saw in Hospital de Obrigo, on a side street, to tend to my feet. Ordered a cafe con leche. Lady brought me scrambled eggs on toast with it. I thought she was feeling sorry for me, so I ordered a second cafe con leche as a thank you. She beings me another small dish. And that is how I discovered that in Castilla & Leon you get a small dish of something for free, the original tapa, with any drink you order. ;0)