This week's book giveaway is in the OO, Patterns, UML and Refactoring forum. We're giving away four copies of Refactoring for Software Design Smells: Managing Technical Debt and have Girish Suryanarayana, Ganesh Samarthyam & Tushar Sharma on-line! See this thread for details.
That said, assuming that you actually have a java.lang.String, then it is impossible to reverse the String without using at least 1 String method. Or at least, one method from some other core Java class (like StringBuffer, or StringReader, etc...) which would call String methods.
You can cheat using the StringBuilder class, but that's not what you want
You can't do this without calling any method of String. That's because it's internals are not accessible without methods. Therefore, there is a minimum of one method to call to make those internals accessible. That method is toCharArray(). You may also want to check out a combination of length() and charAt(int) - those allow you to do the same but without creating a new char (which toCharArray() does).
This sounds like an introductory exercise in recursive programming that I once had. By using length() and charAt(int) as Rob suggested, you can sort of simulate a situation where you are getting characters one at a time. I think this is still in the spirit of the question. Just make sure you read the string from the beginning, and don't use an array or another string to store the characters in reverse.