- I will do the above from Sept. to June every year, except for my total vacation time of 1.5 months (Christmas and Spring) + 3 months (Summer) when I will return home.
Based on this, you have 6 weeks (Christmas and spring) + 12 weeks (summer) of holiday giving you (52-18) weeks = 34 weeks of University per year.
There is a very useful website summarizing the cost of using London buses: https://tfl.gov.uk/fares-and-payments/fares/bus-and-tram
A bus fare in London is £1.50, with a maximum cap of £4.50 per day for bus fares. So you only pay for the first three buses you use each day.
- Minimum weekly travel: I will bus at least 10 times weekly, twice daily for 5 days, from my residence to campus.
So at least £15.00 per week, in term time. Let us say you spend at least £15 per week × 34 weeks every year = £510 per academic year.
- Maximum weekly travel: I will bus at least 20 times weekly, 2x for 5 days, because I lack time to go out more than 2x day.
So that is £4.50 (maximum daily rate) × 5 days = £22.50 per week. (If you pay by Contactless credit/debit card instead of Oyster, this is capped to £21.20.)
So that is £22.50 per week × 34 weeks every year = £765 per academic year.
With your 30% off, you can get an annual bus pass for £592. That is less than your maximal projections, even after the £20 administration fee, but more than the minimum.
- I will travel only within Zone 1. (I lack time to go anywhere beyond!)
I think you are being very pessimistic here. You have your weekends (which you deliberately excluded from your accounting), you have Wednesday afternoons (reserved on the undergraduate timetable for sport), ‘optional’ lectures (in my day a good fraction of the lecture timetable was better spent in absence), a ‘reading week’ if you are doing a humanities subject ... you will do plenty of travelling outside Zone 1.