Please stain a test piece of the same wood that the rung is made out of first, before you stain the rung!
Unfortunately, the fact is that the same species of wood will NOT necessarily stain the same (sorry Steve, but I have to disagree in a friendly way with you here).
First, there are many types of oak and you probably do not know whether the original oak is the same exact species as the new oak.
Second, the grain pattern and color is affected by the growing conditions and by the part of the tree from which the wood is cut. In fact, different parts of the same board may be dramatically different in color and grain (such as heartwood versus sapwood.
Third, the way that the wood is cut froom the tree affects the grain pattern and may affect the color because of how the stain is absorbed (quarter sawn versus flat sawn). This may be less of an issue on a chair rung.
Fourth, the old chair was likely made from virgin timber which has very tight grain and beautiful color that is different from the grain and color in second-growth trees.
So, the answer is to match the stains on test pieces that are sanded the same as the rung. The good news is that it is generally easier to match dark stains such as walnut.