Biased random walk has been studied extensively over the past decade especially in the transport and communication networks communities. The mean first passage time (MFPT) of a biased random walk is an important performance indicator in those domains. While the fundamental matrix approach gives precise solution to MFPT, the computation is expensive and the solution lacks interpretability. Other approaches based on the Mean Field Theory relate MFPT to the node degree alone. However, nodes with the same degree may have very different local weight distribution, which may result in vastly different MFPT. We derive an approximate bound to the MFPT of biased random walk with short relaxation time on complex network where the biases are controlled by arbitrarily assigned node weights. We show that the MFPT of a node in this general case is closely related to not only its node degree, but also its local weight distribution. The MFPTs obtained from computer simulations also agree with the new theoretical analysis. Our result enables fast estimation of MFPT, which is useful especially to differentiate between nodes that have very different local node weight distribution even though they share the same node degrees.