I can't speak specifically about 1901, but Russia did have an embassy in Constantinople (as the city was known in English at the time; it would only become Istanbul in 1930), as did most other major world powers (the US, UK, France, Germany, Austria-Hungary, all did). Further, many countries (Russia included, had consulates across the Ottoman Empire, which until 1912 included: modern Turkey, Albania, Syria, Lebanon, Israel, Jordan, Iraq, Libya, Egypt (de jure; de facto controlled by the UK at that point), and large swaths of Saudi Arabia, as well as parts of Greece and Bulgaria. In particular consulates existed in cities like Aleppo and Damascus (now Syria), Beirut (Lebanon), Baghdad (Iraq), Trabzon (eastern Turkey), and likely some others that I'm missing.

The embassies and consulates were important for the European states in particular due to concessions made by the Ottoman over the previous centuries, which had effectively made European nationals exempt from Ottoman law, and were under the control of their home countries. The Europeans were also working to steadily increase influence in the Ottoman Empire, which was clearly declining as a power (Tsar Alexander II of Russia famously called it the "Sick Man of Europe"). Wars fought with Italy in 1911 and a slew of Balkan countries in 1912-13 saw the loss of what is now Libya, Albania, and nearly all the Balkan territory, while the British were working to further consolidate their control over Egypt (which as noted was still technically an Ottoman territory, but was effectively under British rule since 1882).

To give some more focus on the Russians though, they had major interest in the Ottoman Empire: Istanbul is located at the Bosporus Strait, which connects the Black Sea to the Mediterranean Sea. Russia had, and still has, it's main ports on the Black Sea, with Odessa being the major port of the era (now in Ukraine, and still a major port). A considerable amount of Russian exports, especially grains, were shipped out of Odessa, and had to go through the Bosporus, so they needed to ensure the Ottoman would not block the channel. The Russians also had designs on eastern Anatolia: they had annexed a swath of it in 1878 after their last war with the Ottoman, gaining the cities of Batum (now Batumi, Georgia, a major port then and now), Ardahan, and Kars (both in Turkey). With the advent of oil production in the Caspian Sea, oil was shipped to Batum and sent out to the world, which further meant Russia needed a clear way to the seas.

Both sides also had to worry about the local Armenian population, which at the time was mainly concentrated in the Ottoman side of the border, but also had a considerable number in the Russian Empire. By the turn of the century there had been major attacks on Armenians, including an 1894 massacre in the Ottoman Empire, and so some revolutionary groups were formed (the most famous being the Armenian Revolutionary Federation, known by their Armenian name Dashnaktsuyun, or Dashnaks for short). While the Ottoman and Russians didn't coordinate together on this matter, they did both deal with it, as both were targeted by the Dashnaks.

I can't offer much about the history of diplomacy though, but this may be a bit of an introduction of what you're looking for.