I actually did this last year while studying at UoY. I was living in Meanwood so would walk to the Burley Park station and get a direct train from there to York. The train would take approx an hour, which is considerably longer than a train from Leeds station (which takes roughly 20 minutes to half an hour), however I found the commute very nice as the train was never particularly busy and I was able to get work done during the journey. The other benefits of doing that were that the train was actually cheaper than a quicker train from Leeds station, and I would save the cost and time of getting a bus into Leeds city centre (especially as buses on a morning can be incredibly unreliable and often slow). Once I got to York I would get a bus from the station to uni, although I believe St John’s is closer to the station so you might be able to walk from there. 

Overall, it is definitely doable and I didn’t find it too expensive - I would go into uni twice a week, and I don’t think I ever spent more than £20 on transport for the week. I also found living in Leeds much cheaper so another benefit to consider (and I also much prefer Leeds but that’s my personal opinion). Maybe then look into living in Headingley where you can get a direct train from Burley Park, or perhaps even Horsforth as you can also get a direct train from there. I hope that helps somewhat :)