Would like lodging recommendation in/near Old Town section.
The standard guidebooks all have recommendations for accommodations, and list them by price and service. Trip Advisor is an excellent resource for reviewing various accommodations. I always check for the preponderance of reviews and the latest few just in case something new has happened (good or bad). User-posted photos are good since photos on the official websites always show rooms in the best light.
The grand dame in Bilbao is the Carlton on Plaza Mouya It's about a ten-minute walk to the old town and about the same to the museum. Their cheapest rooms generally run about a hundred and fifty bucks.