From version RC4 you can use makeServletRequest() to generate custom solr queries. See some examples on GitHub.
Here are some code examples:
// Ping example
// class_exists('Apache_Solr_Service') ?: spl_autoload_register('_search_api_solr_autoload'); // make sure that SolrPhpClient class is loaded (use only for <=rc3)
$server = search_api_server_load('my_solr_server'); // CHANGE THIS
$solr = new SearchApiSolrService($server);
$result = $solr->ping();
drupal_set_message(print_r($result, TRUE));
// Simple search query (not fully tested)
// class_exists('Apache_Solr_Service') ?: spl_autoload_register('_search_api_solr_autoload'); // make sure that SolrPhpClient class is loaded (use only for <=rc3)
$server = search_api_server_load('my_solr_server'); // CHANGE THIS
$index = search_api_index_load('my_index'); // CHANGE THIS
$query = new SearchApiQuery($index);
$query->keys('foo');
$query->fields(array('title'));
$solr = new SearchApiSolrService($server);
$result = $solr->search($query);
drupal_set_message(print_r($result, TRUE));
// Custom path example for rc4 and above (not fully tested)
$server = search_api_server_load('my_solr_server');
$uri = SearchApiSolrConnection::LUKE_SERVLET . "?show=schema"; // or any other custom URL
$connection = $server->getSolrConnection(); // or: new SearchApiSolrService($server)
$response = $connection->makeServletRequest($uri);
drupal_set_message(print_r($response, TRUE));
// Example for assembling the right URL
$server = search_api_server_load('my_solr_server');
$url = 'http://' . $server->options['host'] . ':' . $server->options['port'] . $server->options['path'];
Above code tested in hook_init with Drupal 7 and Search API Solr [rc3].
Acquia
If you're using Acquia, you can use Acquia-Search-Proxy which allows you to send custom queries against Acquia Search (Apachesolr + Search API Solr). See also my fork of Acquia-Search-Proxy which include some extra improvements.
Example custom queries via CLI:
PATH_INFO="/admin/ping" php search-proxy.php
PATH_INFO="/admin/luke" QUERY_STRING="show=schema" php search-proxy.php
PATH_INFO="/select" QUERY_STRING="q=foo" php search-proxy.php
PATH_INFO="/admin/file" QUERY_STRING="contentType=text/xml;charset=utf-8&file=schema.xml" php search-proxy.php