1) Newcastle Airport is about 30km from the university. A taxi ride from the airport to Durham costs around £50 and takes about 35 minutes. It is also possible to take the Metro from the airport to Newcastle Central Station and then a train to Durham, which takes about an hour in total. You can buy the Metro ticket to Newcastle Central Station at the airport, and the train ticket from Newcastle to Durham at the train station, although it can also be convenient to buy the train ticket in advance at https://www.nationalrail.co.uk (using "Newcastle" and "Durham" in the "from" and "to" boxes) in case there is a short connection.
2) Another possibility is to fly to other major UK airports and take a train. Examples are London (3 hours), Manchester (3 hours), or Edinburgh (2 hours). Google maps can be used to plan your journey, and train tickets can be bought in advance at https://www.nationalrail.co.uk.