Biological agents navigate complex environments by combining long-term memory of successful actions with short-term suppression of recently visited locations-a capability difficult to replicate in artificial systems. We introduce a neuromorphic navigation system based on a spiking neuron–astrocyte network (SNAN), in which spike-timing-dependent plasticity (STDP) reinforces traversed action sequences on a long timescale, while astrocytic calcium transients suppress moves into recently visited states on a short timescale. This dual-timescale memory biases the agent toward unexplored regions without reward signals or explicit global statistics. In grid worlds where the agent observes only its current position, astrocytic modulation increased goal completion from 45.9% to 99.9% and shortened median routes 5.8-15-fold on commonly solved tasks. Compared with seven classical strategies, including Tremaux's and Tarry's algorithms and tabular Q-learning, SNAN yields the shortest routes among reliable strategies in open and sparse static mazes while storing 2.8-4.6 times fewer variables than memory-based baselines. In dynamic mazes, its decaying memory retains no obsolete passage marks, so SNAN becomes more efficient as the topology changes. After ten learning cycles, STDP makes SNAN the best reliable strategy. The astrocytic modulation mitigates the exploration-exploitation trade-off through local state suppression, a form of working memory we term Topological-Context Memory. To validate hardware feasibility, we map STDP to a memristive VTEAM model and implement a subset of the network on a crossbar array, achieving order-of-magnitude gains in speed per area and energy per decision over CPU implementations. These results establish astrocyte-inspired dual-timescale memory as a compact, hardware-compatible principle for neuromorphic navigation.