I believe you are mistaken in your understanding of asm.js.

First off, from their FAQ

Q. Is asm.js a new language?
A. No, it’s just (a subset of) JavaScript.

And you asked _{clarification added} :

But how is it possible to write one [an asm.js application] in C++

You don’t write an “asm.js application”, rather asm.js is a target¹ to compile your C++ code to.

This article by John Resig provides a number of details that may better explain how asm.js would be used.

Starting with this image:

you can see that asm.js is a translation target of emscripten. Emscripten handles translating the LLVM bytecode into JavaScript, and asm.js is a subset of JavaScript. Staying within asm.js’ restricted subset of JavaScript allows the code to be optimized and run faster.

You also asked:

And if I write the code in C++, then how to use the js API-s

Again, you’re kind of missing the point. Asm.js enables porting existing C/C++ applications into JavaScript so they can be run within a browser. You wouldn’t normally be able to use JS APIs within your C/C++ code, and there’s nothing magical about asm.js to allow that.

If you have a new application to write that needs JS APIs then you should write the application in JS and not futz with trying to write in C++ and then port to JavaScript.

And going back to Resig’s article, there are two key quotes for your question:

the kind of applications that are going to target Asm.js, in the near future, are those that will benefit from the portability of running in a browser but which have a level of complexity in which a direct port to JavaScript would be infeasible

and

As you can probably see from the code above Asm.js isn’t designed to be written by hand. … The most common use case for Asm.js right now is in applications complied from C/C++ to JavaScript. Almost none of these applications interact with the DOM in a meaningful way, beyond using WebGL and the like.

What you might want to consider doing instead is having a JavaScript program that calls the JS APIs that you need along with making calls to the C++ that you compiled to JavaScript. Have a look at this emscripten tutorial to see how to call C++ code from JavaScript.

For some additional research, emscripten has a tutorial that might help you get started with understanding how to take C++ code, run it through LLVM, and then target asm.js.

¹ _{Strictly speaking, that’s not true. The C/C++ code is unaware of what it’s going to be compiled to, so I can’t really call asm.js a target. Another tool (emscripten) takes the LLVM output and then translates to asm.js compliant JavaScript. But I’m going to call it a target because it’s easier to understand.}

Yes, you can write C++ code and compile it to asm.js, using emscripten. I haven’t tried it myself, and I’m not sure how ready this is for prime time. It seems to be good enough to run a bunch of games though.

Here is a tutorial: http://kripken.github.io/emscripten-site/docs/getting_started/Tutorial.html . Looking at the tutorial, it seems quite easy to compile C++ code:

// hello.cpp
#include<stdio.h>

int main() {
  printf("hello, world!n");
  return 1;
}

$ ./emcc tests/hello.cpp -o hello.html

The easiest way would be to use WCPP, a package that lets you import C++ nearly directly into your Node project.

Our C++

// addTwo.cpp 

export int addTwo(int a, int b) {
  return a + b;
}

In the terminal (to compile our C++)

$ wcpp

Our JavaScript

const ourModule = await require('wcpp')('./addTwo.cpp')

console.log(ourModule.addTwo(2, 3))

For more information, see the NPM Package or the Git Repo