In addition to the 5 pm bus
@trecile mentioned, the cuadrabus website shows one at 6:30 and one at 8:00, so unless your plane is really delayed, you should be able to make it to Logroño.
The only thing missing is the schedule for the bus from the airport to the bus station. Looks like it runs every 15 minutes and takes about 20 minutes. 3€ As
@donalomahony says, the bus station is the last stop on the airport bus.
Find out the best way to go to Bilbao from the airport – comparison of travel times & costs for bus, taxi & private transfer plus more practical information.
www.euskoguide.com
Cabs are reasonable, it seems, about 15-20€ from that website, but I would not be surprised if the cab fares rise because of fuel prices.