The railways that support City Centre Stadiums will not be closed because of driverless cars. The railways that will be in serious trouble will be rural routes and branch lines. No really major events take place in rural areas in this country apart from music festivals (Glastonbury etc).
I'm not so sure, at least in the short term after implementation in rural areas it is likely that the charges for use would be higher than rail.
Initially they are likely to be on taxi charges, even if they get to half that (and you've got to remember that in rural areas there's a good chance that such vehicles aren't used as much as urban areas) the cost is still going to be quite high. In today's prices £5.60 is likely to be a 3 mile journey, trains would easily be able to better that even at peak times using turn up and go prices.
In comparison Woking to West Byfleet (3.5 miles) is £4.20. Even an open return is £6.60 (vs £11.20 for the automated vehicle going each way)
£1.40 more for a one off trip and people would probably pay it for the convenience.
If you go further then the taxi rates per mile don't reduce, whilst trains tend to.
However for regular travel, return journeys, longer distance travel and the like and trains would still be used.
That's assuming that prices are half that of taxis currently. Yes, in theory you could see the per mile rate fall below £1.50/mile (current taxi rate is £3/mile for mile 2 onwards), however I'm urban areas prices would likely need to balance so capacity doesn't run out but didn't mean that the vehicles are under used. In rural locations the price needs to reflect that they may not be used as much (at least you might need to have more empty movements to get to the next passenger and/or longer gaps between passengers).