The Golden Temple, Jallianwala Bagh, and the old city are all walkable from each other. For the rest of the city, cycle-rickshaws are the most practical option in the narrow lanes; auto-rickshaws are faster on wider roads.
App-based taxis (Ola, Uber) work reasonably well in Amritsar. For the Wagah Border, shared jeeps and autos run from near the railway station throughout the afternoon — agree on a price before you get in. A return trip by auto-rickshaw to Wagah is roughly 300–400 rupees.