Finn might initially go into denial. He could keep expecting Jake to just pop back up like he always does. Then he'd probably get angry, at himself or at whatever caused Jake's death. Eventually, he'd start to accept it and try to honor Jake's memory by being a better hero.
There's also a possibility that Jake is just an ordinary person in the fanfiction world who has been given this extraordinary ability to imprint. This could make his character development really interesting as he has to deal with the consequences of this sudden and powerful connection to the reader. He might struggle with understanding his new feelings and how to act on them in a world that may not be used to such a thing.
I haven't come across any reports stating that Jake Tapper is penning a novel. His focus seems to be on his journalistic endeavors, such as reporting the news, conducting interviews, and providing analysis. However, it's not impossible for him to venture into novel - writing in the future, but for now, there's no indication of it.