How can I get multisite primary blog (url or path) for current user?

Indeed, get_active_blog_for_user should work.

$blog = get_active_blog_for_user( get_current_user_id() );
$blog_url = $blog->domain... /* or $blog->path, together with $blog->siteurl */

Alternatively:

$blog_id = get_active_blog_for_user( get_current_user_id() )->blog_id;
// note: changed "->userblog_id" to "->blog_id" in row above to make it work.
switch_to_blog( $blog_id ); /* switch context */
$home_url = home_url();
restore_current_blog(); /* back */