Project 21956 is considered a cruiser class.
It is considered to have the firepower of a cruiser... just the same as an OSA class missile corvette was considered to have the firepower of a WWII battleship with its SS-N-2 anti ship missiles.
That doesn't mean the OSA can perform the roles of a WWII battleship, nor does it mean this new destroyer can perform the role of a cruiser.
Putting a nuclear reactor into the 1164s, might as well build new cruiser from scratch
The new reactors will likely last 30 years without refueling, so it might be the case that they are made to be modular so they can be plugged in like a battery. This would mean you could have propulsion pods instead of huge heavy drive shafts and gears. With electric motors and infinite gearing the vessels would become much more manouverable, and more importantly the heavy reactor and all its shielding can be positioned low in the ship near its centre and positioned like ballast.
On a normal vessel the propulsion goes at the end of the drive shaft and you need to place ballast around the ship to balance it so that it is stable in the water. As the vessel consumes fuel and ammo the balance shifts so you need to be able to shift a percentage of ballast to maintain the balance. By using the reactor as ballast you greatly reduce the amount of dead weight on the ship and make it lighter, so it displaces less than it would otherwise displace... making it faster.
There is no point having nuclear powered carriers zipping around the planet at 35knts+ if the cruisers that are supposed to be protecting it can only manage that speed for 4,000NM before they have to refuel... or in the case of the Kirov class 2,000NM before they run out of fuel and their speed drops down to 16Knts on old nuclear only power.
They are building 10 SSBNs and 10 SSNs with nuclear propulsion between now and 2020... they are likely putting another nuke power plant in the Kuznetsov... I think it would be pretty stupid to not put one in the two Kirov and three Slavas they are keeping.
And if nuclear reactors are all that expensive... how could developing a new cruiser from scratch with a nuclear reactor be cheaper? Putting one into an existing vessel will not be without risks but it should be much faster?
Remember I am suggesting a superficial upgrade and then a later much much deeper upgrade on ships that have largely been kept semi operational. This is not the Gorshkov carrier that sat on a dock for 15 years.
21956 is dead for all practical reasons, and Russian navy was never interested in it. What will the new destroyer look like? Maybe scaled up 22350? Or if Garry is right and it ends up being nuclear powered, I imagine the dimensions will be different.
If current practises are anything to go by they will try to cram as much as possible into the ship as they can... Frigates look like destroyers, so can we assume destroyers will have the weapons and sensors of cruisers?