So far as enacting important substantive legislation is concerned, I think the answer is: virtually nothing. This is no reflection on him; rather, it reflects the prevailing political and policy landscape.
So far as enacting legislation with the full expectation of its being vetoed is concerned, that is a matter of his choice and internal Republican legislative politics. I personally see nothing wrong with the Republicans' enacting big bills that they would actually like, but that they know President Obama will veto without serious prospect of override. That can be a legitimate way of framing the Republican side of the 2016 policy debate. But I don't think the Republicans will be able to unite behind responsible tax reform legislation that is genuinely budget-neutral even over the next ten years. This was well shown by the cool reception that Republicans offered Ways and Means Chair Camp's very serious set of business tax reform proposals. While Congressman Ryan would presumably be better situated than Camp to sell such proposals to his own caucus, I doubt that even he could, and also that he would want to.
The really big question is whether Ryan and the Senate Republicans will decide to make major changes in revenue estimating, designed to score tax cut proposals as generally raising revenue, sharply reducing unemployment, etc., even when in fact there is no serious prospect of the proposals actually having the claimed effects. Ryan has been talking this up publicly in aggressive terms. I am willing to credit him with being completely sincere, in the sense of believing that the existing estimating process results in under-measuring the positive effects that he believes enactment of his policy preferences would have. To him, it may therefore seem win-win to push for what is sometimes (on the whole inaccurately) described as more "dynamic" revenue forecasting. Win-win, in the sense that the revenue estimates become both genuinely more accurate and more favorable to his policy views.
But - as the Republicans found in 1994 when they tried to move towards more "dynamic" forecasting - the results they want cannot in fact be yielded by an honest revenue estimating process. And I suspect that leadership figures such as Congressman Ryan, even if they don't currently know this, are about to find out. The conservative or Republican economists who have been named publicly as possible new heads of the Congressional Budget Office are all reputable and honorable people, and they know that an honest, even if "dynamic," revenue estimating process simply can't yield anything close to the numbers that the Republican leadership may not just want, but genuinely believe to be more accurate.
Accordingly, getting the desired numbers, as a regular product of a revised estimating process, would require crossing a Rubicon that at present remains somewhat in the distance. Essentially, it would require dishonestly destroying valuable public institutions that produce information of value to everyone in the process. I am very hopeful that the Republican leadership, once they realize that this is the choice, will recognize that it is neither good policy nor truly in their interest. There is value to a legitimate process, both to provide oneself with better information and to maintain public credibility. And note, of course, that the Democrats may well get back the Senate in 2016, so this really is a two-party game if you look more than two years out into the future.
Again, the Republican leadership in 1995, with Doug Holtz-Eakin playing an important and positive role at the Congressional Budget Office, swiftly came to realize that it made sense to do the right thing (i.e., preserve the legitimacy and honesty of the revenue estimating process). Fingers crossed that it happens again. And while I am not invariably inclined to lean sharply to the most optimistic side when making guesses about the future, I do think there are very good structural and institutional reasons to hope for a positive outcome here.