I doubt that such a built-in function exists in MS Excel. Nevertheless, this problem is a linear regression that is simple enough to solve analytically.
Let us start with
$$\Pi = \sum_{i=1}^{n} \left[y_i - \left(a x_i + b + \frac{c}{x_i}\right)\right]^2 \enspace $$
with $n$ the number of data pairs and $(a,b,c)\equiv (c_1, c_0, c_{-1})$. To find the minimum of this approximation we compute the partial derivatives
$$\frac{\partial \Pi}{\partial a} = 0\\
\frac{\partial \Pi}{\partial b} = 0\\
\frac{\partial \Pi}{\partial c} = 0 \enspace ,$$
obtaining the system of equations
$$\sum x_i y_i = a\sum x_i^2 + b\sum x_i + c\sum 1\\
\sum y_i = a\sum x_i + b\sum 1 + c\sum \frac{1}{x_i}\\
\sum \frac{y_i}{x_i} = a\sum 1 + b\sum \frac{1}{x_i} + c\sum \frac{1}{x_i^2} \enspace .$$
If we simplify the notation with
$$\begin{align}
&m_1 = \sum x_i y_i, &m_2=\sum y_i,\quad &m_3 = \sum\frac{y_i}{x_i},\\
&\gamma_1 = \sum x_i^2, &\gamma_2 = \sum x_i,\quad &\gamma_3 = \sum \frac{1}{x_i},\\
&\gamma_4 = \sum \frac{1}{x_i^2}, & &
\end{align}$$
the system of equations can be written as
$$\begin{pmatrix}{\gamma}_{1} & {\gamma}_{2} & n\\
{\gamma}_{2} & n & {\gamma}_{3}\\
n & {\gamma}_{3} & {\gamma}_{4}\end{pmatrix}
\begin{pmatrix} a\\ b\\ c\end{pmatrix} =
\begin{pmatrix} m_1\\ m_2\\ m_3\end{pmatrix} \enspace .$$
And the solution is given by
$$\begin{align}
a &= \frac{1}{\Delta}[m_3 n^2 + \left(-m_1\gamma_4 - m_2\gamma_3\right)n + \gamma_2\left(m_2\gamma_4 - m_3\gamma_3\right) + m_1\gamma_3^2]\\
b &= \frac{1}{\Delta}[m_2 n^2 + \left(-m_1\gamma_3 - \gamma_2 m_3\right)n + \gamma_1\left(m_3\gamma_3 - m_2\gamma_4\right) + m_1\gamma_2\gamma_4]\\
c &= \frac{1}{\Delta}[m_1 n^2 + \left(-\gamma_1 m_3 - m_2\gamma_2\right)n - m_1\gamma_2\gamma_3 + \gamma_1 m_2\gamma_3 + \gamma_2^2 m_3]
\end{align}$$
with $\Delta = n^3+\left(-\gamma_1\gamma_4 - 2\gamma_2 \gamma_3\right)n + \gamma_2^2 \gamma_4 +\gamma_1 \gamma_3^2$.