Is the word 'branch' correctly used in the following sentence? Or is the correct word some other word like 'field'?
He is a 19-year-old national sportsman. His branch is archery.
No, branch wouldn't be used there.
I also wouldn't use field, but based on comments some readers would find it acceptable.
The most common expression would be "his sport".
If speaking in the context of a multi-sport competition like the Olympics or World Track and Field Championships you could use "his event". This would be more specific than sport. For example two athletes could both compete in the sport of swimming, but one's event could be the 100 meter freestyle and the other's could be the 50 meter butterfly.
You could also use the more general term, "his specialty".
There are also specific nouns that designate competitors in certain sports, such as archer, boxer, cyclist, decathlete, equestrian, fencer, etc., so in your example you could say "he is an archer".