So, how about combining Oxford and Highclere in one day? Is that feasible? Exhausting, yes, I'm sure, but is it doable with public transport?
I would ask my lodging about the best way to get there.
Use the website at www.travelinesw.com, and type in your details. It will involve a number of bus and train journeys, with some walking, and should take between 3 1/2 and 5 hours each way. You will have to leave shortly after 4 to make the return journey the same day. Few locals would make a journey like this on public transport.
Yes it might be a mere 60 miles via the most direct road route (and drivable in under 1.5 hours) but public transport will take you on a long arc route around Oxon and Berks.
I think this would be an exhausting, frustrating day. You won't see much of either place and you'll spend an inordinate amount of time riding around and changing trains and buses. Why not just rent a car for the day? Oxford is only 30 mins or so from Highclere and if you get an early start you can see the house and get a good sense of Oxford. I would give at least two hours to Highclere and then the rest of the day to Oxford. Are you planning on going this summer? Highclere is sold out for the limited days they were offering. A good substitution would be Blenheim Palace, just west of Oxford.