Once the OP knows the condition of the chimney how will you then answer
his question? My interpretation of his question is that he is routing his
stove pipe up an old chimney and the stove pipe will enclose the hot gases
so the condition of the chimney cap and the question of lining seem moot.
I'd be concerned about the flammable dirt which may be in the chimney. I'd
also be concerned about poor maintenance, in the future, of the stove pipe,
since "out of sight is out of mind".