The commanding officer of a Gorshkov frigates is a captain 1rank ( capitano di vascello in italian ranks, equivalent to army colonel, instead of a captain 2 rank) probably also because they are the most modern and capable surface ships in Russian navy and because there are not many larger ships in service.
If I am not mistaken the commanding officer of an American Burke class destroyer is a commander (equivalent of a captain second rank in Russian navy, capitano di fregata), but that is possibly because they have much more large surface ships than Russia and they do not want too many people in the highest rank below flag officers.
Eventually it is possible than if in the future they will have more gorshkov M, very large hamphibious ships (the >8k displacement modified Ivan gren) and helicopter carriers, some of the basic 22350 could be commanded by a captain second rank.
Are 22800 Corvettes and 22160 patrol ships commanded by a captain 3rd rank?
The Admiral Gorshkov is commanded by a Captain 1st Rank, but checking now reveals the Admiral Kasatonov (also a Gorshkov class) is commanded by a Captain 2nd Rank. So it's a Captain 2nd Rank station.
The Karakurts seem to be commanded by Captain 2nd and 3rd Ranks, so it's a Captain 3rd Rank station. It would be classed as a 3rd-rank ship class so it makes sense
The 22160 patrol ships are commanded by Captain 3rd Ranks and Captain-Lieutenants so it's a Captain-Lieutenant position. Despite the fact that these ships have a greater displacement than the Karakurts. Quite surprising actually. Captain-Lieutenant is a rank you can make it to by your late 20s.
You can see the classification of ship ranks in the Russian Navy here, although the list doesn't seem to be complete: