As to the first part of your question asking people if they write letters by hand, my answer is "No, I don't." Frankly I have been using my computer for so long (I still remember working on a PC with one removable disk drive for programs & data and using a singly floppy disk whose capacity was measured in kilobytes and running on a Microsoft DOS version 1 operating system) that my handwriting has declined to the point where even I have a difficult time reading it. So, other than signing my name and scribbling notes in meetings, I rarely write with pen and ink.
However, as to the second part of your question where you ask if it is believable to have a character who writes hand-written letters this can be believable. According to some articles in the Wall Street Journal and other publications, there is a growing niche market to service those who do write hand-written letters and notes. These places produce and sell elegant stationary, note cards, etc. and their business seems to be growing as a number of people like the idea of sending and receiving hand-written letters in an age when most writing is digital.
Good luck with your book.