Train, hmm I suggest the bus
Slightly longer but they usually stop right in front of most major hotels. Save you a lot of pain from hauling the luggage up and down stairs of the train station. Cheap about 200NT or so (100nt about 3 bucks I think)
Cards should work if you are cirrus or plus network
How much you need depends on where you stay and what you eat and do. Decent hotels cost about 3000NT per day and plus food etc, that would come out to 5000NT per day. Of course you could go budget and spend about 2000NT per day. There are a lot of love motels in TPE, cheap but not crappy. Some are VERY nicely furnished. But of course you have to bear with your neighbors noise levels. My friend who is a country manager for a large MNC was hosted in a love motel for 3 months before they sorted out his apartment. Only complain he has is he gets pretty frustrated living there.
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