Traditional TSV (Through Silicon Via) table in vertically partially connected three-Dimensional Network-on-Chip (3D NoC) only stores TSV address information, which easily causes network congestion. In order to solve this problem, a record table architecture was proposed. The record table stored not only the nearest four TSV addresses to the router, but also the input-buffer occupancy and fault information of the corresponding router. Based on the record table, a novel adaptive unicast routing algorithm for the shortest transmission path was proposed. Firstly, the coordinates of current node and destination node were calculated to determine the transmission mode of packets. Secondly, by using the proposed algorithm, whether the transmission path was faulty and got information of buffer occupancy was obtained simultaneously. Finally, the optimal transmission port was determined and the packets were transmitted to the neighboring router. The experimental results under two network sizes show that the proposed algorithm has obvious advantages in average delay and throughput compared with Elevator-First algorithm. Additionally, the rates of losing packet under Random model and Shuffle traffic model are 25.5% and 29.5% respectively when the network fault rate is 50%.