Let we have algebraic equation on one variable. Which methods (exept Sturm's theorem and Descartes' rule) exist to find real roots of equation (or real positive)?
Well, it depends what you mean by finding. Computer algebra systems commonly use isolation methods (approximations) that are based on an improved Uspensky's algorithm (by Rouillier and Zimmerman), but that's based off Descartes's rule of signs.
On the other hand, you can encode a real root by specifying the signs of all derivatives of the polynomial at the root. Thom's lemma guarantees that there cannot be more than one root satisfying all the sign conditions. Sturm's theorem helps you find which conditions are met. This does not give you any approximation of the root, but surprisingly enough allows you to compute symbolically with them. More details about this can be found on the book by Basu-Pollack-Roy freely available here.
I realize both of these options are refinements of the things you didn't want to talk about... I don't know of any other methods though (or if I do, they're escaping me right now).