[Noriffik]

Вместо

public IHttpActionResult GetProduct(int id)
        {
            var person = persons.FirstOrDefault((p) => p.Id == id);
            if (person == null) return NotFound();
            return Ok(person);
        }

возвращай JSON:

public JsonResult GetProduct(int id)
{
            var person = persons.FirstOrDefault(p => p.Id == id);
            return Json(person, JsonRequestBehavior.AllowGet);
        }