Was there a way to accelerate the process of reforms compared to OTL so Russia performs better in WW1 to at least establish a Russian mandate in former Ottoman territory?
With a benefit of a hindsight, it is rather easy to say that most of the
army reforms of 1905 - 12 could be conducted prior to the RJW (which, BTW, could be easily avoided, sparing a lot of expenses). This definitely goes for the organizational aspects: geography-based mobilization system, retiring older officers and generals, strengthening the engineering services, improving officers' and non-coms' training, perhaps earlier emphasis on construction of the strategic (as opposite to the commercial) railroads, etc. Probably the same goes for improving a weaponry. In OTL during the RJW most of the Russian units in the Far East were still armed with Berdan rifles even if a newer Mosin–Nagant M-1891 rifle was already widely available (3,800,000 had been produced by the time war started). Can't tell for sure if there were some technological problems which would prevent development of the bigger calibers few years prior to OTL but 76 mm gun model 1900 was already available by the time of the RJW and the problem was mostly with the ammunition: according to Count Ignatiev, there was something of an unhealthy pre-war addiction to shrapnel at the expense of the explosive shells. Then, if we are moving closer to the WWI, there was an early ammunition crisis eventually resolved by moving a lot of the shell production to the small tool shops; at least some preliminary measures could be taken for an earlier deployment of that resource.
Then, with a benefit of a hindsight, we can tell that a big problem was almost complete absence of the canning industry/refrigerators - this was causing growing supply problems both at the front and in the rear (the meat rations in a fighting army had been consistently decreasing, see Brusilov's memoirs); I'm not sure if the strategic reserves of grain big enough to supply cities and army for a prolonged period of time could be created (probably not) but surely state collection of the food during the war could be better organized (even without going all the way to the looting practices of the Reds during the RCW).
The same goes for the better organization of the military hospitals at least on a level of a prepared infrastructure and cadres.
The
big problem was a backwardness and low efficiency of the Russian agriculture but, short of the fantastic projects, it is not clear how this could be remedied: it was unrealistic to supply all peasants with the modern agricultural equipment at the state's expense and forcefully introduce the more advanced agricultural practices. As a result, mass mobilization would inevitably result in a noticeably lower agricultural production. However, even without going into the excesses of RCW and WWII, a better than OTL structure of collecting the needed supplies at the time of war could be created.
The reforms promoting industrialization (to a certain degree was done by Witte) could be done more forcefully minimizing Russian dependency upon "technological" imports like car/plane engines and even ignitions. As far as aviation was involved, in OTL Russian government was anything but helpful in promoting its broad development so by just removing the restriction, it could get the prepared civilian cadres of pilots before the war. With the exception of motors, planes of the WWI did not involve any sophisticated technology and could be reasonably easily produced even in the tool shops (I'm not talking about the heavy bomber, "Ylia Muromets", planes constructed by Sikorsky).
Car industry also was lagging behind: in 1914 Russia had only 10,000 cars (Britain 245,000, France 245,000, Germany 57,000) and not a single car-building plant (most of the domestic cars had been built in small numbers on the train factory in Riga, only 451 passenger car and few trucks had been built there; there were few other places producing cars in small numbers) but in 1915 government sponsored creation of 5 specialized car-producing companies (none of which seemingly was functional by 1916). Total production during the war amounted to 20,000 assembled from the imported components. Surely, the measures could be taken well before 1915.
The navy was a problem by 2 main reasons: 1st, this was a time of fast developments (Dreadnought was launched in 1906) and due to the Russian industrial backwardness construction of the capital ships presented big technical challenges. However, the tasks of the Russian navy had been recognized as defensive both on the Black and Baltic Sea which should make things somewhat easier.
On a political level, it is more or less taken for granted that earlier introduction of constitution and increased power of Duma (all the way to having a responsible government) would resolve most of the problems plaguing Russia in 1914 - 17. However, it should be remembered that Duma was not an equivalent of a magic wand and that it was filled by the people fighting for political power and mostly completely incompetent in any issue that truly mattered in the terms of preparation to war or conduct of a war. A brief experience of the Provisional Government can serve as an illustration: things were bad when it came to power and they managed to make them catastrophic in just few months just by a pure incompetence. OTOH, it is probably fair to say that existing system was not adequate for the task either, especially on the top levels. Nicholas II was incompetent (and rather lazy) ruler and absolutely unsuited for a role of a supreme commander. Nicholas Nicholaevich did have a military education but clearly was not up to the task either, pre-war plans were inadequate and the list is going on.
So the only good policy would be to stay out of any serious war and to forget about the absolutely unnecessary ideas like occupation of a big part of the Ottoman empire.
Edit:
The fundamental problem for the Russian Empire was a technological backwardness which in OTL resulted in a need of huge purchases abroad, which simply could not always be done:
1st, Russian gold abroad had to cover payments by the existing debts and France put severe restrictions on the ability to spend gold stored in the French banks abroad (Russia was making big orders in the US). After much bargaining with France and Britain Russia got saddled with the new loans big part of which had been channeled to paying interest on the existing loans and only fraction spent on the military purchases.
2nd, neither France nor Britain had a surplus production capable of covering the Russian needs without damaging their own interests. And when during the war your industry can't produce enough rifles, ammunition and even infantry helmets, you are looking for a big trouble.
So one fundamental thing that needed to be done was intensive industrialization up to a level allowing to cover most of the war time needs by the domestic production. But I'm afraid that the time was lost: reign of Alexander II (when the process could happen) saw the low import tariffs which were killing a possibility of domestic industrial development. Situation changed during the reign of Alexander III but the process was not going fast enough. Then, of course, there would be a need for the huge numbers of the industrial workers but for having a pool of the "raw" cadres (which could be taught the new professions at a workplace) you'd need an impoverishment of a big part of the Russian peasantry all the way to a complete inability to remain in their villages. Can you imagine the scope of the social problems related to such a scenario? All "progressives" and all bleeding hearts at Zemstvos bemoaning sufferings of the poor peasants and trying to stop the process, peasant revolts, etc. And, of course, there would be a need for the capital and expertise to build up Russian heavy industry within few years.