There are daily minibuses leaving Kilwa Masoko for Lindi and Dar leaving early in the morning - buy your ticket in advance and especially if going to Dar try and sit as close to the front as possible.
It is possible to get a daladala / shared taxi / private taxi to Nangurukuru and try to pick up one of the larger buses going either north or south that pass by between 10am and 2pm but be warned that these buses are often full and you may find that you cannot get on one.