Hope my English will be sufficient for you to understand.

Mongol invasion had a pretty big effect on Russia from the military standpoint (asides from being one of two most destructive enemy incursions in Russian history with amount of destruction comparable only to that of German invasion during World War Two). Many scholars are rather uncertain, though, of the extent of this impact and what exact part of it (direct or indirect) changed overall method of conducting warfare among Russian principalities, whether these changes happened abruptly or over a long period and whether or not they were some kind of deliberate response towards Mongolian invasion or successive evolution of Russian arms under different circumstances. Amount of firsthand literary sources is very limited and archeological findings from earlier periods are very scarce. Thus, I will try to put it as mildly as I can without going into the field of wild theories.

In case of adapting to Mongolian technologies and tactics during the invasion proper the answer would be no, Russia didn't try to copy, analyze or successively utilize Mongolian strengths against themselves mostly because of how fast and all-consuming this invasion happened. The exact change from "old" to the "new" forms of Russian military structure is also hard to decipher due to many similarities they already had with nomadic people.

Russian military forces during this period were comparable to those utilized by Western European armies with some exotic additions. First of all, as people deeply interconnected with steppes who had to beat back multiple nomadic incursions over centuries they put great emphasis on ranged horse combat which was achieved by both training their own feudal retinue and enlisting steppe foederati to serve as mobile archer units, so called sharpshooters ("streltsy", not to be mistaken with musketeer corps of Ivan IV, these are whole other story). Role of highly armoured heavy cavalry chargers would be entrusted to knyaz (prince) and his loyal guards who would observe battlefield from their stavka (field command post) and anticipate decisive moment to attack.

Second, Russian forces had little to know knowledge of operating siege equipment and conducted sieges by primarily starving the defenders out, so Mongolian stone throwing machinery came as a big surprise to them.

Third, Russia during entirety of its' medieval history had little to no dedicated infantry forces like Western European militiamen or mercenary crossbowmen and many Russian military engagements of this time were almost entirely made out of cavalry hosts changing blows with each other. There were exceptions like Sudovaya Rat' ("Naval warhost") but they were deeply specialized and, frankly, just exceptions to the rule. This was caused by both abundance and knowledge of horse warfare coupled with enormous distances which Russian armies often had to travel. Considering their enemies were mostly nomads which utilized lightning speed as their main weapon of choice and scarcely populated countyside with not near as many people as, say, medieval France or Spain (so not enough castles to deterr enemy movement or pin them down) it was only logical for infantry to be exceedingly rare. Russian medieval states were no Romans, they had no way of effectively training large amount of infantry which could be useful against horsemen so they completely disregarded footmen as mostly irrelevant.

The problem here was that Mongols did all this but better - they had much better tactics so it was easy for them to trick Russian and Kipchak army into the trap at the battle of Kalka in 1223 and decisively destroy it. Main reason why Russians were so eager to fall for this was their rather favourable military history against nomads - they thought they already knew everything steppe horsemen could throw at them and misunderstood complex Mongolian maneuvers as a sign of weakness. Curiously enough, some Russian warlords were rather cautious and advised against haste decisions but their words weren't heard by anybody. Allied forces were completely destroyed, but this didn't bother principalities which weren't present at the Kalka river enough as they later received news of Mongolian hosts being ambushed and defeated by Bulgars. This surely put many princes into more comfortable position as they had many victories scord against Bulgars and thus enemy who suffered a misfortune against them would surely be of no threat. Words uttered by survivors from Kalka fell on deaf ears of course.

Conquest of Russia took only three years and resulted in complete defeat of armies of principalities affected by it and elimination of many important cities. Russian forces were usually easily outmaneuvered and outshot by better trained and disciplined Mongol hosts time and again, and amount of destruction laid by enemy horde was unparalleled with any previous war Russians have ever faced. Notable comments on opposing forces from Russian chronicles state that Mongols were outstanding archers and masters of siege warfare, with stone throwing catapults being mentioned multiple times. With either death or enslavement of many veteran warriors and craftsmen Russian military technologies take a great step back: Russian equipment of XIV century is considered to be of more crude quality and variation than that of earlier epochs as arts of weapon and armour making took very painful blows with downfall of many trading and manufacturing centers.

So one of the short term effects of Mongol invasion would be extreme weakening of Russian military for years to come. Some principalities, such as Novgorod and Pskov, of course, were largelly unaffected by enemy incursions and continued to rely on their old war knowledge, but heavy burden of Mongols (and their successors, the Golden Horde) demanding both military and material help from their newly found vassals was very tangible to the entirety of Russia.

You rightly asked how pre-modern society would go about copying enemy technology, because it wouldn't. Russians seemingly didn't try to copy Mongol army overnight and mainly stayed true to their old paths slightly modifying theim in the process. It happened partially because simply adopting new bows and raising new horses wouldn't turn vassal states into military powerhouse comparable with that of the Empire as utilizing them correctly was actually even more important. Aside from siege equipment it is usually perceived that Mongol discipline and superiour tactics had much more weight than their archery, with Roman Khrapachevsky, one of the most knowledgeable modern Russian researchers of this topic, arguing that downfall of Rus' happened wholly because of better organization and logistics of Mongols but not because of their better weaponry or armour. With misfortunes mentioned above it was impossible for surviving Russians to quickly adopt or replicate Mongolian art of warfare as they had no previous experience of building unified and highly disciplined fighting force and now also possessed no resource base to build it upon. It also would've been very naive to expect their new overlords to simply allow their weak vassals to rebuild themselves and grow back stronger than ever so Golden Horde during entirety of its' lordship over Rus' was very careful in managing different principalities and often directing them against each other because of this exact reason. Small scale victories such as Daniil Romanovich defeating army of Mongol commander Kuremsa during his punitive expedition in 1255 didn't change the fact that Russian lands were too disorganized and barren and couldn't resist their steppe opponents on larger scale.