I don't see why not, but with some creativity.
Here is the current schedule from the (one?) company doing the route.
http://www.autobuseslaunion.com/PDF/Bilbao-Logrono.pdf
You would have to take the 8pm bus which would get you into Logrono 9:30 or10:00, too late to jump in a taxi and arrive in Viana before albergues close.
But... This gives you time to make it to central Bilbao, go enjoy a few tapas while stretching your legs and head to the bus station for the 8pm bus.
You could then take a taxi to Viana and stay in a pension or the beautiful hotel at the end of the main street in Viana . They would give you a room after 10pm.
You could also spend the night in a pension/hostal in Logrono, drop off your bags in the am at the albergue, or ask the hotel to hold them for you, bus or taxi to Viana in the morning and enjoy the short and easy walk from Viana to Logrono, just over 10km.
This is especially possible if you are not flying in on a transatlantic flight, which would be unusal into Bilbao, and if you did and then took an internal flight and then the bus, that will be your day to get over jetlag, napping on internal flight and bus ride.