Kursk Station is the Moscow terminal for trains from central Russia (Kursk and Nizhny Novgorod), eastern Ukraine (Donetsk and Kharkiv), and some Sapsan high-speed trains from St. Petersburg. The Peking Hotel is around 8 km from the station.
By taxi
It should take around 30 minutes to reach the hotel by car, although the trip can be much longer at peak times. If you are willing to negotiate with a driver on arrival, a taxi should cost no more than $25. If you prefer, you can book a transfer with a driver to meet your train here.
By metro
Alternatively, take the metro from Kurskaya Station, and travel one stop west on the dark-blue line to Ploshchad Revolyutsii. There change to the dark-green line. Travel two stops north to Mayakovskaya, which is just across Triumfalnaya Ploshchad from the Hotel Peking.