Hmm, you are looking at a €100 - 130 cab fare assuming the driver doesn't want "return" money. You'll miss the 1700 train to Logrono assuming that's your flight landing time. You might do ok with a bus to Vitoria and a connection to Estella. Or spend a night in Bilbao and get the early morning train to Logrono and decide whether to turn East or West when you get there.
Paddy, it is quite a distance from Bilbao to Estella. I usually get the 6pm bus from the station in Bilbao to the station in Pamplona. That gets in to Pamplona just before 8pm. You could stay there overnight and get a bus to Estella next morning, they leave from the bus station and it takes less than an hour...
so, airport shuttle from Bilbao costs 3 euro and leaves every 20 minutes. Coach from Bus station in Bilbao to Pamplona at 6pm, around 15 euro, and then next morning the bus to Estella is about 4 euro. So that is 44 euro more or less. Another option - have you heard of blablacar? Great invention! You just need nerves of steel in case nobody is driving from Bilbao to Estella at a time and date that suits you!
Ok, here is another option: book with omnio, so you can print your tickets ahead of time and you just have to find the bus stand: Bilbao to Pamplona at 6pm, getting in just before 8pm. From same station, same level, bus to Estella at 8.30pm and it takes about an hour. Howzat? I attach the screenshot of the timetable from Pamplona to Estella. If I can be of further help, pm me.