Well, you will get to matriculate (get dressed up in 'subfusc' - suit, white bow tie, and gown - and be talked at whilst sitting in uncomfortable seats in the Sheldonian theatre) and graduate (similar but with more walking in and out, bowing, and changing of gowns/hoods) so that is some tradition ticked off straight away! If you do a course with exams, you will also get the pleasure of doing said exams in said subfusc.
Outside of that, it depends how much you want to throw yourself into being stereotypical 'Oxford'. The opportunities are there if you want them - white tie balls, getting invited to dinner at as many colleges as possible etc. On the face of it, it sounds like you might want to apply to one of the older colleges - although be aware the graduate accommodation is often in the modern complexes of even the oldest colleges. You should do some research into the colleges though as, for example, some colleges do regular dining where you are expected to wear gowns (fairly normal dinner but dressed up) whereas others have largely dispensed with it except for an occasional formal dinner (often with fancier food). Some colleges also have more ridiculous traditions than others - tortoise racing (Corpus Christi I think), walking backwards drinking port when the clocks go back to stop a rip in the time space continuum (Merton), choir singing from top of tower on May Day morning (Magdalen - although that is a huge city event that loads of people attend) to name a few. Or you might decide actually you hate all the traditions and want to ignore as much of it as you can which is also perfectly doable (except for the matriculation and exams bit which you have to do - though you can skip graduating in person if you want).
As for not having done undergraduate at Oxford, I don't know the exact figures, but a lot of graduate students come to Oxford from elsewhere (again, look at individual colleges as the size of graduate body varies) and, certainly of my undergraduate cohort, only a few stayed on (and a lot of those were medics doing their part 2 or people on 4 year undergraduate masters courses who get to join the graduate body in their 4th year). The ones who have been to Oxford as undergraduates are more up to speed on the jargon and idiosyncrasies of Oxford and probably come across as more confident in their surroundings for the first week or so but they are not somehow 'superior'.