can I stay in an albergue that is on a different route?

This is one of those things that will be totally up to the discretion of the hospitalero. When I was in Oviedo last summer, there were two college kids from the US who had walked from Porto to Santiago and were now traveling around Spain by bus. He let them stay in the albergue (for at least two nights) because there was room, but of course walkers had preference. In other albergues, you cannot stay unless you plan to walk the next day. In still others, you cannot stay unless you have walked in.

Unless you are planning to visit Oviedo, though (totally worth it), I think it might be more of a hassle to get into town, find the albergue, and then get to the bus station to get to Leon, than it would be to just go directly from the Asturias airport to Leon.