In AE (American), I would say "as much as possible".
I think "far" is correct here, but it is confusing because "as far as possible" can also mean "for the longest possible distance", and public transportation is about vehicles that travel a distance. For example:
I want to travel to my daughter's home. I will take the train as far as possible, then a taxi for the last 3 miles.''
I would avoid "far" only for that reason. In other sentences, "as far as possible" is fine.