I was in Taiwan working teaching but after 90 dyas left my school (and Taiwan) and didnt fullfil my contract, should I get a new passport before I go back? will I have problems getting another ARC?
I doubt they will care. If you have any record here, they go by your name and date of birth, not your passport number.
The only problem I forsee would be when you apply for a job with a new school. They may ask for a reference from your old school.
Also, the visa office in your home country will automatically do a search to see if you have a record in Taiwan. You will come up as having been in Taiwan before.