Kursk Station, the Moscow terminal for trains from central Russia (Kursk and Nizhny Novgorod), eastern Ukraine (Donetsk and Kharkiv), and some Sapsan high-speed services from St. Petersburg, is roughly 2km south of the Hilton Moscow Leningradskaya.
By taxi
Travelling by taxi from Kursk Station to the Hilton Moscow Leningradskaya Hotel should take no more than 20 minutes. Taxis will cost around $15. You can book a transfer with a driver to meet your train here.
By metro
From Kurskaya Metro Station (connected to the mainline station concourse) travel one stop anticlockwise on the brown circle line to Komsomolskaya, which is less than five minutes' walk from the Hilton Moscow Leningradskaya.