Per the question, are there even barbers in Taiwan, or only hair stylists? How much can I expect to pay for a shave, if it is possible to get one?
There's a great review of what sounds like a quality one over on forumosa, titled "Best. Barbershop. Ever."
It includes directions to the shop, and points out that they did:
It doesn't say specifically that they'll shave a beard, but unless you're some sort of sasquatch that grows a beard while getting dressed in the morning, you'll likely be fine (even then, they sound professional enough to handle it).
Cost: $400 Taiwanese, or around US$13 (present day exchange rate). A shave is $150 TWD.
Perhaps this is a bit old, but you can show up to the barber I used to use. They did beards with a straight blade. It is perhaps a bit out of the way. It is about a 3-4 minute walk from Tamsui Station at the end of the red line. A hair cut was 400NTD, I'm not sure how much a shave is. I will say that they always did a good job with my hair and have experience cutting foreign hair styles.
2nd Floor, 251新北市淡水區博愛街4號 沙茶羊肉, Taiwan