Is it possible to install new module without ftp in drupal 6?
If all directories are writable, you should be able to enable PHP module, create a node with PHP filter, and in that node use:

file_put_contents ( 'path_to_modules/mymodule/mymodule.info', 'name = example
description = Demonstrates how to write info file.
core = 6.x');


The same for module file.

Preferably do that with rudimentary module that will allow you to upload and extract zip / tar.gz module files as provided by Drupal.org, and then remove PHP filtered node and disable PHP module.