If Boo were to write a letter to Jem, I imagine Boo would ask about how school was going for him and what sports he liked. He would probably ask politely about the well-being of the household. Once establishing a common interest, the two might discuss sports or goings-on around the town. He might ask about Dill and just what the boys had been doing in front of his house that summer. Finally, Boo might speak about the fire at Miss Maudie's house that cold winter night. I'm sure he would inquire after Jem's arm that was injured.
Jem, I doubt, would ask Boo about any of his exploits from the past. Atticus had let both Dill and Jem know the topic was closed without further consequences. Jem desperately wanted to please his father. He might ask if Boo had mended his overalls that night and thank him for it. If Boo showed any interest in common to Jem, the two might discuss that. Jem might even compliment him on his soap carving abilities as demonstrated by the two figures he and Scout found in the knothole.