Best way to get to Josephine Butler college.Watch
If you still want to take the taxi to Jo Butler, then I suggest that you try and find some people who are going to Durham and convince them to take the taxi with you (thus splitting up the bill).
Otherwise, you can take the Metro (£4?) from the airport to Newcastle Central Station and then take the train to Durham (around £6) and then the taxi to Jo Butler (another £6 maybe/no one wants to walk all the way from the train station to Jo Butler).
Hope this helps