Trishkerry - I did this journey twice last year.
The local bus service from Biarritz airport (bus stop just outside the main entrance to the left) is run by
www.chronoplus.eu . Depending on when you arrive and whether it's a weekend or not there are generally 2-3 buses an hour (Ligne C) to Biarritz train station (which is only 2km away and a five-minute journey). There's also an occasional bus service direct to Hendaye, which takes a while as it meanders a lot.
From Biarritz station there is a regular train service to Hendaye - again it depends which day of the week or hour you are travelling - check here
http://uk.voyages-sncf.com/train-tickets/book-your-train-tickets . It's about a 25 minute journey. When you come out of Hendaye station you keep walking the direction the train would have taken you if it had continued. Keep straight ahead for the bridge over to Spain (about 3 minutes). Just as the road starts to rise, you'll see there's a metro station on your r/h side and if you are short of time you can catch this into central Irun. See this map
http://www.enterat.com/al-dia/2012/10/09/web-metro-donostialdea/ . The third stop is Colon which is about 5 minutes from the albergue.
But I recommend walking if you have time. Leave the traffic that detours left and keep going straight ahead over the historic old bridge which now just carries pedestrians. As you come onto the bridge you should see your first yellow arrow....
On the other side there's an old style border town greeting with bars, dodgy hotels and trinket shops (a bit like arriving in Mexico..). You just keep to the road going straight ahead up into the town which takes about 15-20 mins. If you are planning to stay at the albergue, you turn right when you reach the main drag/high street called Colon Ibilbidea and keep walking for 10 mins (past the Colon metro) and over the railway bridge where you turn sharp right and downhill and pick up the signs to the albergue which is about 150m away...
Thanks for asking the question - I really enjoyed retravelling that it in my head!
cheers, tom