Aside from the minor edits, I think it sounds very natural.The task: A boy you met last summer on an english course has written to say that he will be visiting your contry on a holiday next month. He has asked for advise on what places he and his family should wisit. Reply to his letter.
Thanks for your letter. It was a lovely surprise to hear from you after all this time. I'm sorry I haven't written for ages, but I've been really busy ["real busy" is more informal]
theesethese days. I still study a lot and I have to work to get by somehow.
I'm not sure what you mean by "I have to work to get by somehow." Do you mean " I have to work hard to get by."?
I'm so excited that you and your family are planning to come to Russia. I promise you'll love it! It is ["It's" is more informal]
very beasvery beautiful in the winter. The snow is everywhere and everything is shining. I hope this winter will be snowy.
You asked for advice on what places to visit. I think you should
definitlydefinitely see the Kremlin and TheRed Square. I've heard that foreigners are fond of itthem. You should also walk along the so-called old Moscow [area/district/street?]. It looks fantastic in the evening with all thesethe lights. And then you might visit me and my family. My mom and I will cook something special for you.
I'd better stop now and get on with my studying. I hope to hear from you soon.
"I'd better stop now" sounds very abrupt. You might just say something like -- "Well, I've got a lot of studying to do." or just leave it out. You really don't need to give a reason for not writing a longer letter.