The airport buses pick up from the Westgate (a 2 minute walk from the Premier Inn), but there's no drop-off there on the way into Oxford. It's about the same walk from the St Aldates stop and the Gloucester Green stop (the two nearest ones that inbound coaches stop at), but there's no taxi rank at St Aldates

The stop at the Westgate going back to the airport is S4, on Speedwell street roughly behind John Lewis - https://goo.gl/maps/gJ3cSpu615vnqQio6