score:3
Travelling from Taipei to Chiayi, and then directly to Alishan National Park is already a very long journey, the 7329 bus journey from Chiayi HSR to Alishan is over 2 hours in itself. Adding Fenqihu into the trip will definitely be stretching your time significantly - you will be spending more time travelling than standing still, although the Forest Railway museum at Fenqihu is a good little place to visit.
Incidentally, there is also a park dedicated to Forest Railway locomotives in Chiayi, slightly south west of North Gate Station - Alishan Forest Railway Garage Park.
As for the Forest Railway itself, you can buy tickets on the day (Source, in Chinese but this information translated by my wife) but you may well be disappointed as this service is very popular, especially at weekends. That said, according to this website (English at the bottom of the page) you can order tickets online up to 15 days in advance on the Alishan Railway & Cultural Heritage website, however it appears to be mostly in Chinese so you may require some help. I'd definitely try and make the extra effort to make the trip on the train though, it's a lovely route with gorgeous scenery.
Going back to the buses: Looking at the timetable you posted in the comments and assuming you take the free shuttle service from the HSR to the TRA station, your first opportunity to get from Chiayi TRA to Fenqihu is the 0935 service on line B (7322), which will take just under 2 hours assuming it sticks to timetable. Then there is a 1250 service from Fenqihu to Alishan (7322a) which should take about an hour and a quarter. This'll give you a couple of hours at Alishan National Park (as well as the highest Post Office and 7Eleven in the country) before catching the 1710 service back to Chiayi.
(All photos taken by myself in October 2014)