Bilbao? Please don't rush off and grab the first train/bus out of town. Ease into the timelessness and culture the Camino offers and visit the Guggenheim Museum of Frank Gehry no need to go in and the nearby Basque Museum do go in. Get your Credenciial stamped at the cathedral as a souvenir and then hop a train/bus for Pamplona.
I have used Bilbao to bus to Burgos and on another occasion to bus to Pamplona. However, I did stay overnight in Bilbao on both occasions. As its a mid day flight out of Dublin, it was more convenient for me to stay the night and get an early morning bus to Pamplona and start walking as soon as I arrived up to Cizur Menor. In fact, from Bilbao there are direct buses to Pamplona, Logrono and Burgos. If you choose to stay, try the Pil Pil hostel in the San Mames district. About 5 minute from the bus station and in the shadow of the football stadium. It is good clean and includes breakfast. As it is on the Camino del Norte, it is used by pilgrims on that route so you can get a Sello on your credencials. Also, about 20 minutes or so walk from the Guggenheim if you wish to visit there before starting your camino. The bus ride is a couple of hours. Dont know how long a journey it is from Barcelona but would imagine it would be a fair bit longer. Plus, if you miss the bus there is another not long after as it is a regular service
Do check out Pil Pil hostel, but you should get food fairly early. There are a couple of bar/restaurants just round the corner but stop serving food about 5 or 6. Personally, I brought some Cup a Soups, got some bread and that did me until breakfast. There is a decent kitchen and as I said, breakfast is included. If you are travelling on the Irish Passport Card, this hostel was the only place I had a problem but they eventually recognised the word 'passport'
It is possible to get the 6pm bus to Pamplona from the bus station in Bilbao, after the Dublin flight that is due in to airport around 4.15. The bus from the airport is easy to catch, as tpmchugh says. I think the price is 1.75 but I forget, really. I got my last ticket online as I had chosen to travel on one of the worst days of the year - the Friday of San Fermin. Actually, given that you have to get in behind the building facing the buses, to get to the desk to buy your ticket, even though you have to pay about three euro to the company rome2rio suggests, it is worth doing it so you just have to show up at the bus rank and keep your eyes peeled for the bus to Pamplona. Seats are assigned, so no hassle. Then you can sleep in Pamplona and head off early in the morning to climb el Perdón. An alternative to staying over in Bilbao. You have choices to make!