This sentence is pretty good and passable, in my opinion. A few things that could improve it a bit: You might want to put a subject in the question... How long did it take YOU to get used to (the) cold weather? I don't think "the" is technically required in this case, but it does sound better with "the" before cold weather. Someone could say "I don't like cold weather" and "the" isn't required, but since St. Petersburg is a specific place, using "the" is better.

You got most of the sentence correct; you just need to put a preposition between who it was that you got used to (mother-in-law) and how long it took you to get used to her (3 months). As teacher Brass indicated, both "in" and "after" can work, in this case. I prefer usage of the word "after" because the longer version of the sentence could read "after 3 months' time had passed". If the sentence were negative, the word "for" would fit better: I wasn't (/didn't get) used to living with my mother-in-law for 3 months. I would say: I got used to living with my mother-in-law AFTER 3 months. Picking the proper preposition can be difficult, even for native speakers. Remember that nothing can be substituted for practice, so just keep practicing! Don't forget to hyphenate "mother-in-law".