Some take coins and others take phone cards. You can purchase phone cards at any convenience store. They are also sold in vending machines. Sometimes these vending machines are located right in or near the public telephone booths. It is probably best to place international calls from your hotel, so hotel staff can assist you.
Taxis are expensive in Japan.
However, there are excellent alternatives, such as airport buses and high speed trains. Japanese people you encounter in business and at hotels often speak excellent English, but many people outside of these settings are not skilled in English. It is extremely helpful to learn a handful of Japanese words, phrases, and symbols, especially the common courtesies. The Japanese will appreciate any effort you make, however simple, to speak with them in their own language.