It's fairly easy to live without a phone in Taiwan. It's not like China. Most payments in Taiwan are made with cash, EasyCard (or equivalent, depending on the city. It's a prepaid metro card you can use at some stores) or a credit card. Mobile payments tied to your identity (eg. WeChat) are not the norm. You can also use an EasyCard without providing a phone number but then you cannot rent bicycles or recover the funds if you lose your card.