On that map, there are two distinct fare zones, with a buffer zone in the middle (denoted by the green line for Freeway 1 and National Highway 62). If you are travelling within the orange zone at either end of the map, your fare will be NT$15. Within these sections EasyCard is best used, though pay attention to the sign that states whether you need to tap in on boarding or tap out on leaving.
That said, as of 1st July 2019 buses within the Taipei City and New Taipei areas are moving to a tap-in-tap-out system. In theory this should mean you are charged the correct fare when leaving the bus.
Going by the fact there are five "Fare Increase" sections along the route, I would imagine that each extra stop denoted by a red dollar symbol incurs an extra NT$15, up to the maximum fare for the route of NT$90. In your case this would be a moot point as going from Beimen to Juifen will incur the maximum fare anyway. Note in the middle where it says the NT$90 fare applies to:
Banqiao & Taipei City to Ruifang & Jioufen
As for how to pay the fare for long distance travel, according to my wife you would probably need to ask the driver for a ticket to this destination. However that's the only part I cannot advise on with complete certainty. I've only ever travelled on short-distance bus services within Tamsui, and last time I visited Ruifang I took the TRA train.
Finally, for departure times, you're best just getting to the stop and looking at the real-time passenger information display. In a busy urban area like Taipei and all the traffic that goes along with it, it's almost impossible to timetable bus services to minutes of the hour. Instead, the frequency information states:
- Operational hours: Weekdays 0600-2100, Weekends 0700-2100
- Weekday peak: 30 to 40 minutes
- Weekday off-peak: 40 to 60 minutes
- Weekend peak: 30 to 40 minutes
So take a look at the departure board and if there's not one due for a while then hole up in your nearest Family Mart or 7Eleven for a bit.