Edit 2015-12-14 – adapt to use elements package
The idea is the same as in the original answer, see below. The bohr package doesn't provide \atomicnumber any more. It is now provided by the elements package. The latter also provides \saveatomicnumber<cs>{<element>} which makes the macro a little bit easier:
\saveatomicnumber\@tmp{#1}
and then
\the\numexpr#2-\@tmp\relax
for the number of neutrons.
\documentclass[a4paper]{article}
\usepackage{chemformula}
\usepackage{elements}
\usepackage{tikz}
\makeatletter
\newcommand*\mychemistry[2]{%
\saveatomicnumber\@tmp{#1}%
\ch{^{#2}_{\atomicnumber{#1}}#1}
\tikz[baseline]{
\node[anchor=base,shape=circle,draw,inner sep=2pt]
{+\atomicnumber{#1}};
}
(%
\atomicnumber{#1} p,
\the\numexpr#2-\@tmp\relax
n%
)%
}
\makeatother
\begin{document}
\mychemistry{O}{18}
\end{document}
Earlier answer using the bohr package
An answer to the edited question: \atomicnumber is not expandable and as a consequence not usable in \numexpr ... \relax. However, if you're certain that you only input the element symbol and never the element name you can use the internal command. For the element F this is \@bohr@atom@number@f, for Ne it's \@bohr@atom@number@ne and so on.
In order to use it with uppercase atomic symbols as input you need to \lowercase{...} the input. \lowercase again is not expandable but you can simply surround the \setcounter part with it. Actually, the counter isn't even needed since it is only used to determine the number of neutrons. It is possible to use \numexpr directly by preceding it with \the. In order to build the needed macro name we can use the primitives \csname...\endcsname or LaTeX's wrapper \@nameuse{...}, both of which are expandable. In the command this would then be something like \csname @bohr@atom@number@#1\endcsname or \@nameuse{@bohr@atom@number@#1}.
Putting this together you can get the number of neutrons with
\lowercase{\the\numexpr#2-\csname @bohr@atom@number@#1\endcsname\relax}
or
\lowercase{\the\numexpr#2-\@nameuse{@bohr@atom@number@#1}\relax}
Adapting you example, I added indentation and comment chars where needed but I also left a space between the different parts since it looked more readable to me this way. I also omitted to name the \node since alignment with the baseline can also be achieved with anchor=base as option to the node.
\documentclass[a4paper]{article}
\usepackage{chemformula}% the package that provides `\ch` and which is loaded by `chemmacros'
\usepackage{bohr}
\usepackage{tikz}
\newcommand*\mychemistry[2]{%
\ch{^{#2}_{\atomicnumber{#1}}#1}
\tikz[baseline]{
\node[anchor=base,shape=circle,draw,inner sep=2pt]
{+\atomicnumber{#1}};
}
(%
\atomicnumber{#1} p,
\lowercase{\the\numexpr#2-\csname @bohr@atom@number@#1\endcsname\relax}
n%
)%
}
\begin{document}
\mychemistry{O}{18}
\end{document}

BTW: if you use chemmacros already you can also easily output protons as p+ and neutrons as n0:
\documentclass[a4paper]{article}
\usepackage{chemmacros}
\usepackage{bohr}
\usepackage{tikz}
\newcommand*\mychemistry[2]{%
\ch{^{#2}_{\atomicnumber{#1}}#1}
\tikz[baseline]{
\node[anchor=base,shape=circle,draw,inner sep=2pt]
{+\atomicnumber{#1}};
}
(%
\atomicnumber{#1} \prt, % <<<<<<<<<
\lowercase{\the\numexpr#2-\csname @bohr@atom@number@#1\endcsname\relax}
\ntr % <<<<<<<<<
)%
}
\begin{document}
\mychemistry{O}{18}
\end{document}

\prt actually just is the equivalent of \chcpd{p+} and \ntr is \chcpd{n^0} so chemmacros isn't really needed and chemformula would suffice.
\prescriptfrommathtoolspackage, for instance, would work), and to draw the circle around an element (tikzpackage would do). There are lots of answers here in this site for doing so. However, may be you want an easy way to type that automatically… perhaps adding more info to the question would help. What input would you expect? A minimal compilable example would be great to invite us to help. – Manuel Jul 18 '14 at 10:59\documentclassto\end{document}of something, just so we can copy that code and start working from there rather than typing it ourselves, is always welcome. – Manuel Jul 18 '14 at 11:07bohrprovides\atomicnumber{<element>}(e.g.\atomicnumber{F}gives9) – cgnieder Jul 18 '14 at 13:52