I don't think it's a question if you can play those last Beethoven sonatas depending studying music at the university or not, but the fact that you started playing piano late. Just like language you just learn it better when you're young. The natural understanding of music and it's meanings and cadans. But... The positive thing is that you are a mature person and you'll grasp things faster than kids, so you're process will be faster going through the stages of learning piano.
I do think it is a good investment if you have a teacher spend 15-30 minutes every week specifically for theory and solfege (ear training) and general music - if you really want to play the Schubert and Beethoven at its greatist some time. And it ís a great investment, because it's extraordinary to get into this music and know more about it and figure it out through the months studying it!
This can be done by your teacher, or you could take a teacher privately especially for that (maybe a student from the university, I'm sure they'd love to do that). Talk about music, listen to a lot of them (concerts, CDs), do some eartraining games, composing. That will take care of the learning about the background of music to understand it better, no need for the University of you don't want that.
I think you will be able to play everything. If you listen to a lot of classical music, have it as a normal thing in your life and you're musically gifted I'm sure you'll be able to "understand" the more difficult music and it's depth, and play Schubert and Beethoven with much care.
(I'm just talking about the musicality in music, because technically you can play anything you want, all the Beethoven sonatas, we're just talking about the extra fun it gives you to actually emotionally play it satisfactory for yourself)
Just out of curiosity... I have a 17 year old student who also just started a half year ago. What kind of pieces are you playing, how much have you learned already in the past 6 months?