Let $f(x) = g(x) - 8x$, and then $g(x)$ is also a quintic polynomial with its leading coefficient being 1. And there is
$$g(n) = f(n) - 8n = 0, \quad n = 1, 2, \dots, 5.$$
Hence, $g(x)$ has 5 real roots, namely, $1, 2, \dots, 5$. Therefore,
$$g(x) = (x-1)(x-2)\cdots(x-5),$$
and thus,
$$f(x) = (x-1)(x-2)\cdots(x-5) + 8x.$$
Consequently, the coefficient of the term of degree 1 of $f(x)$ is
$$\left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}\right) \cdot 5! + 8 = 282.$$