From RI to Richibucto, the fastest way is probably by car. Take the I95 all the way up to Houlton, crossing over at Woodstock, NB. Take Hwy 2 across to Moncton, transfer to Hwy 15, then go north along Hwy 11 to Richibucto. I know this involves some backtracking, but my husband and I once took Hwy 1 through St. Andrews/Bangor and found it longer due to speed limits & going through so many towns. His cousin, who lives in NH, likewise always takes the I95 to Woodstock and then down Hwy 2 to get to Sussex because she finds the Interstate faster.
If you want a more interesting way to go, take the Cat across from Portland Maine to Yarmouth, NS. It's 5 hours to cross the Gulf of Maine, after which you can take Hwy 101 north to the Digby Ferry and cross over to St. John, follow Hwy 1 to Moncton, etc.