Yes..... Rest of the code in the doStuff() will available to all the threads. This is the good adavantage of declaring syncronized block ,over declaring method as syncronized. Because , It keeps the entire code in the method in lock mode.
Using the connection polling name , you can create the database connection. You can use datasource to get the connection. If you are not clear refer the document "Getting Jdbc connection using DataSource " in Google Search.