Feel free to share your experience here or on another new thread if you have time to type it up! I would love to hear about it.
Correct, depending on where you are situated in the city, there's buses from a few places around the city to get to the airport.
From my knowledge buses leave from Banqiao Station, Taipei Main Station, Taipei City Hall Station and Song Shan Airport. Luckily, all of these places also have MRT stations. So you can save a little money on the cab fare, but would have to lug the bike box up and down the MRT stations.
There's also the Airport MRT. I've never taken it myself, but I hear it's about the same price as a bus ticket from Taipei Main station to Taoyuan airport.
Lots of options, just a matter of where you're located in Taipei!