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I agree with the above response. If you are a woman arriving that late, I recommend staying at a hotel near the airport. I have used Ibis and TRYP. Both have shuttles. When my daughter arrived solo late at night, I had her take a taxi to the hotel. Even, if you are male travelling solo...
As noted in earlier messages, the bus works well. It leaves from T4 terminal and goes straight to Burgos (I did this twice and once it made a stop along the way). The arrival in Burgos is only a few blocks from the Cathedral and the Camino.
The carrier is Alsa. I bought my ticket online...
I'm making that same trip this Wednesday and taking the bus. I made the trip from Burgos to the airport last fall. Alsa picks you up at T-4 and the Burgos bus station is about two blocks from the cathedral. As noted in another reply, the trip is about two and a half hours.
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