I don't think this is the correct PoD to accomplish this, although I'm just going to wing it here. Correct me if I'm wrong. Perhaps, instead, Russia manages to succeed a bit more and creates a stalemate. I'm not sure how, but they manage to cause the Serbs and Bulgarians to actually revolt against the Ottomans and a stalemate is reached in the Balkans. Eventually, Bulgaria gets its independence (probably with its smaller borders). Russia doesn't gain territory anywhere, but they don't lose anything.
Fast forward to a likely inevitable Russo-Turkish war, and from a stronger initial position Russia's gain are even greater. This is when Russia would march into Constantinople before any intervention can be organized. As long as the situation on the ground remains under Russian control, even if any treaty is not recognized by other powers at first, the distracted world powers might not intervene for long enough.
Either that, or Russia is rebuked again. They gain more in Armenia and San Stefano Treaty Bulgaria might come into effect, but no one recognizes Constantinople has been conquered and war is threatened. Maybe or maybe not Russia backs down here, but they still eye it.
If WW1 evolves... similarly (Germany's rise occurs similarly, threatening Great Britain's control of the sea irrevocably pushes them towards France and Russia), and a stronger Russia comes out (the revolution would have to be averted, of course) of the war, they'd probably end up with Constantinople de facto as their spoils of war. The treaty of Sevres was drawn up after the war, so there won't necessarily be a an International zone.
This would involve a lot of parallelism, and the prospect of an allied Bulgaria does change things a bit, but the overall larger picture shouldn't be impacted so much as to prevent this from happening. Russia just needs to do a little better in each of those wars.
Now, as to whether Russia would make it its capital? Probably a ceremonial of some sort. For it to be a capital, it would probably encompass all of eastern Thrace and the Mamara region just so it wouldn't be an isolated spit of land. (And, in this situation, Greece likely has Ionia to "make up" in it losing Thrace to Bulgaria/Russia, so it even more friendly territory nearby. I wouldn't doubt that the Tsar would set up some kind of semipermanent residence though. ...It'd make it interesting if it eventually lead to a multicapital system of some kind, even if it just means that the Tsar spends x amount of time in the various palaces throughout the nation.