As the title says, when did the Varangian rulers become culturally more Slavic? I presume that among the elites there was cultural borrowing from Byzantium too, so I am not claiming a Viking-Slavic binary distinction.
If, then when, did the mother tongue of the Kievan rulers switch from Old Norse to Slavic?
The mainstream historical view, delivered by historian Francis Donald Logan: "in 839, the Rus were Swedes; in 1043 the Rus were Slavs". That's perhaps an overly cautious view, but in between, it does get kind of fuzzy. Let's see if we can narrow it down some.
There have been two big problems dealing with this question out in recent years. One is the fact that we don't have a lot of first-hand written records of the period in question (requiring us to resort to archeology and linguistics). The other has been that area identity issues got caught up first in Nazi/anti-Nazi propaganda, and later in nationalist propaganda.
The (Norse) Varangians were largely traders and raiders, setting up their towns on good river sites, and extracting tribute from neighboring Slavic and Finnish tribes, who likely always outnumbered them.
This is very similar to the situation ongoing at the same time on the other end of the Norse world, where French-speaking Normans were in charge of England, while most of their subject people were Anglo-Saxons. Given that both kingdoms reached the same cultural result, it seems reasonable to expect a similar path, where the ruling-class was eventually absorbed, with the Kings being perhaps the last Norse-speakers to become completely assimilated to the local language (and culture).
The rulership line to look at here is the house of Rurik. There is no real (mainstream) debate that the Rus founder (b. 872) was an Old East Norse speaking Varangian prince. He would have been capable of conversing comfortably with the Danes living at that time in England.
Rurik's grandson Sviatoslav (b. 943), was the first Rurik with a name of Slavic origin, and he worshiped the old Slavic gods, so it appears we may already be there. Sviatoslav did however spend a lot of his time collecting "tribute" from neighboring East Slavic tribes, which seems to indicate the two groups still felt some cultural separation.
His son Vladamir (b. 958?) apparently still had enough cultural ties to the Norse homeland to enable him to go back there and recruit mercenaries to take the throne back from his brother, but everything else about the guy looks Slavic. He had a Slavic name, worshiped Slavic gods, and after assuming the throne mostly only needed to campaign against non-Old East Slavic territory, such as the Poles, Volga Bulgars, Croats, and of course Byzantines. There was mention of conquering the East Slavic(?) Radimiches, and a rebellion among the Baltic Yotvingians. (The fact that we're thinking in terms of "rebellion" rather than "refusing tribute" is pretty telling here)
By the time of Vladamir, large numbers of his (mostly Slavic) subjects had converted to Christianity. Vladamir followed suit in 987, dragging the rest of the country with him. Vladamir's son Yaroslav is accepted by pretty much everyone as fully slavic*, although he did ally himself with Sweden, and married a Swedish princess.
So the answer is that the first fully Old East Slavic Rus' ruler was likely Vladamir, and possibly his father Sviatoslav, but almost certainly his son Yaroslav could be considered fully Slavic.
* - Given that Yaroslav, along with his next 3 successors and his grandfather had names ending in "-slav", the temptation is to say "It's right there in the name!". However, the etymology of the "-slav" ending (Old Slavic for "Glorious") is completely separate from the English word "Slavic". Still, you may notice that's an Old Slavic word, which makes it still a decent point.
As TED points out, there is a lot of modern "scholarship" on the topic that attempts to backdate Slavic identity to Oleg or even Rurik. But the best sources we have for that time period remain the Primary Chronicle and observations of Arab, Latin, and Byzantine emissaries, so these are the sources I cite:
The chronology of Varangian Rus rulers is well-established: Rurik was succeeded by Oleg (Helgi) who took Kiev, then Rurik's son Igor (Ingar).
Things get a little bit interesting when Igor is killed by Drevlians and his wife Olga (Helga) takes the throne. While Helga is usually considered to be of Varangian origin, some scholarship places her as Bolgar :
Игорѧ жє ожєни [Ѻльгъ] въ Българѣхъ, поѧтъ жє за нєго кнѧжну Ѻлгу
Igor was married [by Oleg] among the Bulgarians, to the duchess Olga
Either way, their son Svyatoslav notably has a Slavic name (Svyato - holy or arguably Sveto - bright, Slava - glory). He famously spurned his mother's conversion to Christianity (source: Tale of Bygone Years aka the Primary Chronicle):
Он же не послушался матери, продолжая жить по языческим обычаям.
He did not listen to his mother, continuing to live by pagan tradition.
However, the chronicle does not mention which pagan tradition he followed (Norse or Slavic), as the monk writing it would not have given the distinction much thought.
Also named in the Slavic fashion, Svyatoslav's son Vladimir (Vlad - rule/possession, mir - world) was raised by Dobrynia, a Volhynian Slav. Vladimir is famous for baptizing the Rus people, thereby cutting any remaining traditional links to Varangian culture. Notably, Scandinavian names disappear from the chronicle past this point. The heroes and notables of Rus, from Vladimir onwards, are Orthodox Slavs rather than Norse Varangians.
