I don't think your Japanese will be much help, except for interacting with Japanese tourists. Where there are guides or directions in Japanese, there will almost certainly be guides or directions in English. Your kanji may help with Taiwanese signs or Korean newspaper headlines, but Japanese is not related to Mandarin and distantly if at all to Korean, aside from a handful of loanwords.
Japanese is not widely spoken in either South Korea or Taiwan, and has not been for decades. English, on the other hand, is taught in both countries starting in elementary school— not to say English is necessarily widely or well spoken. Nevertheless, you are far more likely to find people with some knowledge of it, especially in cities and in customer-facing occupations, as opposed to Japanese.
I doubt you would encounter open hostility for merely speaking Japanese. These are modern and well-educated countries, whose citizens can distinguish people from politics (especially in cities and customer-facing occupations). Second, while Japanese colonialism was traumatic, most of the population in both countries has been born since 1945 and has no firsthand experience of it. Third, there is a cultural resistance to open displays of hostility of any kind, especially in front of foreigners. Of course, if you walk into a department store and address the sales clerk in Japanese, she may think a silly Westerner has forgotten what country he's in.