While there are rules on the carriage of bicycles on Russian trains, from experience we know that how they are enforced can sometimes depend on the provodnitsa who is in charge of your carriage.
Generally, though, it is ok to carry bikes on-board. Though obviously the smaller they are, the better. And if they fold up, then better still.
You will be required to store the bike in the baggage car of the train, which is usually located between the train engine itself and the passenger carriages. There will be an attendant here to check in your bike, who will give you a receipt or ticket, which you must show when you leave the train so you can retrieve your bike.
It is not possible to purchase space in the baggage car when buying your rail tickets, as the cost will depend on the size and weight of the bike, so must be done at the station itself.
If you have booked exclusive use of a compartment, you may be able to store the bike in your compartment with you, but this is completely at the discretion of the provodnitsa of your carriage. If you plan on trying this, be sure to arrive at your train in plenty of time to have this discussion. This will give you time to return to the station to purchase a baggage ticket if you are unsuccessful.