That particular bound doesn't hold if I recall correctly, but the following bound does:

**Theorem :** (C. Huneke and K.-i. Watanabe) *The multiplicity of a $d$-dimensional variety with rational singularities and embedding dimension $n$ is at most*
$${n - 1 \choose d - 1}.$$

In the case of a surface, this reduces to the bound you mentioned above. This is an unpublished result of Huneke and Watanabe (currently under review). You could certainly ask them for a preprint.

**EDIT:** My previous answer said that this was a conjecture, and that Huneke and Watanabe proved something related to this, but I wasn't sure if they actually proved this. It turns out that they did indeed prove this, and I got their permission to post that this was indeed a theorem of theirs.