Quick answer: Taipei runs about EUR 80 per day mid-range - noticeably less if you eat local and walk. Cheapest window: October-December and March-April - mild and relatively dry; summer is hot with typhoon risk.
What things actually cost
MRT ride
0.70 EUR
bubble tea
1.80 EUR
night market snack
1.50 EUR
beef noodle soup
5.50 EUR
Taipei 101 observatory ticket
18 EUR
public hot spring bath in Beitou
1.20 EUR
Eat well for little
Xiao long bao at Din Tai Fung (original Xinyi Road branch) - a basket plus sides for about 10-14 EUR
Beef noodle soup at Lin Dong Fang or Yong Kang Beef Noodle - the national dish, about 5-7 EUR
Pepper buns (hu jiao bing) at Raohe Night Market's entrance stall - about 1.80 EUR
Stinky tofu at any night market - braver than it smells, about 2 EUR
Bubble tea from Chun Shui Tang, which claims to have invented it - about 2 EUR
Money mistakes to avoid
Eating or drinking on the MRT - it is fined, and locals take the rule seriously
Doing only night markets - the old town around Longshan Temple and the teahouses of Maokong show a different city
Carrying no cash - night market stalls and many small eateries do not take foreign cards
Skipping the day trips - Jiufen and the northeast coast are the region's highlight and easy without a tour
Transport without the tourist tax
The Taoyuan Airport MRT reaches Taipei Main Station in 36-50 min for about 4.50 EUR. In town, buy an EasyCard and use the immaculate MRT; most rides cost 0.60-1 EUR and YouBike shared bicycles fill the gaps.