Go to the hotel, and try to check in anyway. Most Russian Hotels operate the 'Russian booking system' whereby you pay for the 24 hours, meaning that you can check in late, check out late etc. Most long-distance train and air travel is overnight so it is quite standard to check in and out of hotels at this time.

To pass some time, start off with Red Square and the usual sights - they will be quiet in the early morning. Cafes and bars will be open in the centre - most cafes are now 24-hour places with DJs spinning tunes even in the early morning. Breakfast menus will include fried eggs, cheese, kolbasa etc.

Have fun, carry your passport with you at all times and do not let the police anywhere near your money regardless of how insistent they may be.
Christian Court