Nizhny Novgorod has a metro system with two lines, useful for getting from the main train station to the city centre and to the lower town. Buses and minibuses (marshrutki) fill out the network. Taxis are widely available and app-based services (Yandex) are cheap by Western European standards.
The city's geography works against easy walking between upper and lower districts — the bluff is steep and the distances are longer than they look on a map. The Kremlin and Bolshaya Pokrovskaya Street are walkable from each other; getting from there down to the Oka embankment or Rozhdestvenskaya Street involves a significant descent. The Chkalov Stairs are the most scenic way down to the Volga bank.
For crossing to the far side of the Volga, the cableway to Bor is the most interesting option, though there are also road bridges.