The answer is yes. Just use geodesic normal (also known as exponential) co-ordinates. If you have a book or two on Riemannian geometry, just look for that or a discussion of the exponential map.
[ADDITIONAL COMMENT] For a 2-dimensional metric, it's a nice exercise to figure all of this out using Jacobi fields. In fact, in my opinion, the best way to work with and understand the exponential map (which is a natural parameterization of all radial geodesics emanating from a point) is via Jacobi fields and the Jacobi equation. For example, it leads to an easy proof of a standard result, namely that the coefficients of the Taylor series of the exponential map at the origin contain only the Riemann curvature tensor and its covariant derivatives. The answer to your question follows from this theorem.