1. Download the javascript file from the release page. The js file is optimized, about 732KB in size (190KB gzipped) and can format about 20000 lines of C++ code per second. Small impacts on memory usage.
2. Include the file in the web page where you want to use AStyle to format codes. For Example:

<script language="javascript" src="AStyle_v1.0.js"></script>

3. Call the formatter somewhere else in your js code.

High Level API:

/**
 * @param code - Source code, string
 * @param style - allman / bsd / break / java / attach / kr
 *                / stroustrup / whitesmith / vtk / ratliff / gnu / linux
 *                / horstmann / 1tbs / google / mozilla / pico / lisp
 *                @see http://astyle.sourceforge.net/astyle.html#_Basic_Brace_Styles
 * @param indent - count of a layer of indent, between 2 and 20.
 * @param lang - 0 for C/CPP/Objective-C, 1 for C#, 2 for Java
 * @returns Formatted Code.
 *
 * Supports: C, C++, Java, C#, Objective-C
 */
function beautify(code, style, indent, lang);

Low Level API:

astyle_instance.ccall("process", "string", ["string", "string", "string", "string"], [param1, param2, param3, param4])

Where param1..4 are command-line parameters. Accepts the same as those in AStyle itself.

Work on the AStyle Core is not discussed here.
The modified part and also the export logic is at the very bottom of astyle_main.cpp, where you can preprocess the parameters you want to pass from javascript code according to Emscripten and AStyle specifications.
Building the code is quite easy. Just get Emscripten ready and run this command (params can be changed to your flavor):

<path to python2 executable> <path to emcc.py> --emscripten-cxx ASFormatter.cpp ASLocalizer.cpp ASResource.cpp astyle_main.cpp ASBeautifier.cpp ASEnhancer.cpp -std=c++11 -O3 -s INLINING_LIMIT=1 --memory-init-file 0 -s MODULARIZE=1 -s EXPORT_NAME='AStyle'

The release is built under windows with clang as the frontend.