Note: I wouldn't put this on a public host for now, there are probably bugs!

If you go the other how to access PHP code from dart client, how to implement it?

You can run Dart itself on a server using Dart.io, what's the need for Apache exactly? If you're doing something relatively simple perhaps its sufficient for your needs to write your own webserver in Dart itself, for example serving static pages ought not be too difficult. Depending on what you're using from PHP, it might not be beyond possibility to provide that functionality yourself.

It sounds like the OP wants to use a Dart client with a PHP server and communicate between them, which would mean that the server really is irrelevant.

In the release article, there is this disclaimer:

Note: I wouldn't put this on a public host for now, there are probably bugs!

Also, the last commit was 2 years ago, so I don't think it's actively maintained.

That being said, if you clone it and build it and install it for your Apache host, you can likely use Dart in Apache:

https://github.com/sam-mccall/mod_dart

If you go the other how to access PHP code from dart client, how to implement it?

Easiest is to decide on an intermediate representation, such as XML or JSON.
Then write POST handlers in PHP, and POST to your Apache/PHP server from your Dart client.

Then a more specific question. Is it possible to Dart to send POST request to a server, and then accept the result at the same place in the script and not updating the page?

yes it is.

http://stackoverflow.com/questions/14558210/send-data-between-php-and-dart