Hi, I'm a current student at Royal Holloway - the buses in Egham are mainly Abellio buses, and a single ticket from Egham station to the campus is £1 as swarfliam mentioned earlier. However, I personally use the Campus Service (operated by the University) which has a frequency of every 25-30 minutes and waits outside Egham station and takes you directly to Founder's. It costs £1 too for a single ticket, but considering the public buses are quite delayed a lot of the time, it is a much better option.
I don't really recommend a bus pass at all, if you are living on campus. You'll probably be going into Egham town centre 1-2 times a week depending on what you need, you're better off paying the £2 for a return ticket each time - it's cheaper than a weekly pass. It takes 15-20 minutes to walk to Egham town centre from Uni, and considering on your way back you're walking
up the hill, I would recommend taking the bus.
Oyster Cards are not valid in Egham - the last valid stop is Feltham, which is three stations away. I would strongly recommend a 16-25 Railcard though - I have one myself and it gives you massive savings on your train tickets, definitely worth the investment if you are visiting Manchester often! And while you're at it, invest in the 3-year one, it's cheaper than renewing the 1-year card two times over!
Hope that helps and feel free to ask any further questions
Good luck at Uni!