ON A WEEKDAY, for a foldable bike, it must be folded and bagged. The bag can be a cheap 10NT black garbage bag, multiple 10NT black garbage bags, an old IKEA comforter or a cardboard box. I do not think the MRT employees care what covers it, as long as it's completely covered. You are not allowed to wheel the bike through the station, must be carried.
To my knowledge, a bike bag, will not cover the bike that's not folded or disassembled.
Wherever you're buying your bike, ask the owner to order a bike bag for you. Done and done.
Personally, I don't take the MRT unless the ride starts at a location in which I have to ride over an hour. Sometimes I have no choice bc the meet up is so early, trains aren't even running yet.
If it's a weekday, it could be worth it since you're not paying extra for the bike, but anytime during rush hour and you're lugging around a bike in a box/bag could be a major pain in the ass.
Bike on the MRT during the weekend, no hassle of having to bag the bike. Of course, they make you pay 80NT flat fare for not having to bag it. Something you might want to consider is if you're not taking it early in the morning or , the MRT can get packed with people going on day trips!
What's the reason you want to rely on the MRT though? Crazy traffic? Not familiar with roads? Zombie-like pedestrians? All of the above are very valid reasons and I understand completely, but let me know and I'd be happy to give you advice.
My suggestion is, get used to the bike at the nearby riverside park and/or get familiar with the roads in Taipei that have bike paths. Once you know your away around, it could help and you do not have to rely on the MRT to get around.
All in all, I think the MRT should be a last resort. Say, if you're just too damn tired to ride back home or your friends are meeting you at a location that you have no idea how to get to, the MRT is there.