Can anyone recommend a budget hotel or hostel that is in the center close to metro station
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Look into 途中 台北 國際青年旅舍 / On My Way Taipei Hostel. It's a nice spot and isn't expensive.
Maybe try Airbnb too?
There are a lot of hostel and hotel in Taipei. Of course, hostel is much cheaper than hotel. It really depends on what kind of quality you are looking for. I think you can use booking.com to get some ideas. I was working in the hostel which called the Green World Hostel, and I like how their workers work with guests. They provide you breakfast and sometime they will have a tour to go around Taipei. There is one metro station down by the hostel and lots of good food around it. Hope I answer your question