I need to go from Taipei to Nuannan tomorrow on the "Fuxing/Shuttle 4178"
The website is down and if it doesn't have an English or Spanish version, it wouldn't help me anyway. (http://www.railway.gov.tw)
Not really. What I did last month, on the way out, was to go to Taipei station the evening before, and I bought a ticket for the next morning. On my way back, since I didn't know exactly at what time I would finish my meeting, I just showed up at Zuoying and bought a ticket for the next train.
No need indeed.
If you travel for short distance, saying just city-to-city level like Taipei - Nuannuan(in Keelung) or Taipei-Banqiao, you can take Fuxing(Local Train), which is the shuttle-level train type in Taiwan.
If you travel for long distance like Taipei - Taichung, or Kaohsiung - Hualien. It's highly recommended to book tickets 2 week ago before your trip, with the train type Tze-Chiang, including general Tze-Chiang, Puyuma and Taroko, and train type Chu-Kuang. These are for longer distance transports. (It would take way longer time if you get on Fuxing train.)