Yes, at Oxford you can choose to only study Greek. I'm a current Classics student @ Ox so obviously only have experience of that. However, as I understand it, there is a greater element of prepared set text for the Oxford course compared to the Cambridge course (e.g. the 2nd year Iliad module at Cambridge has much less translation compared to Mods at Oxford - though the Oxford mods format will be changing by the time you arrive!).
I would suggest closely looking at the course papers. The Oxford Finals/Greats (final exams) papers are generally more structured around an author, genre, or time period, e.g. '146-46 BC', 'didactic', or 'Hellenistic Art and Archaeology'. There is only one Reception paper available for finals at Oxford. The Cambridge papers - as far as I remember and can see on the website - seem a bit more thematic e.g. 'slavery in the Greek and Roman World', 'Classics Live', 'Visual Narratives'. You also have a bit more flexibility with the Tripos system I believe.
Cambridge current modules:
https://www.classics.cam.ac.uk/student-information/undergraduate-studentsOxford current modules (Mods is changing soon, probably by the time you would arrive):
https://www.classics.ox.ac.uk/classicsHowever, everyone at Oxford does a 4 year course if it appeals to you to be in the same cohort as everyone those you started with.
Basically, both are really good courses, very well-respected for Classics, just a little different in their approach. Don't know if that helps at all, but happy to answer any questions about Oxford. You might also want to think about whether you prefer Oxford or Cambridge as a place to live!