I would like to know if there's a way to refresh the Auditing log programmatically, because we need it to see in real time which the last accessed document was within a Document Library.
What do you mean by update? Change the data?
If you just need to view the data, some code like this has worked for me:
Then you just loop through the auditCol and add it to a dataset and bind to a datagrid for output.
You may want to make sure you are setup to audit the events you are looking to capture. Here is a good overview blog post: http://sharepointknowledgebase.blogspot.com/2010/09/sharepoint2010-site-collection-auditing.html