That's not really how it works. The short story is that Duke has a fair amount of flexibility in how it distributes indirect costs derived from research (which funds are paid in arrears) that are supplied by government grants, and indirect costs alone account for several hundred million dollars per year received by Duke. Given that the funds are generally received about a quarter after they are incurred, how to absorb cuts is mostly a matter of discretion. Traditionally, athletics have mostly been immune to perturbations in the research enterprise, since the athletics department raises a lot of funds on its own. However, on the scale we see here, if Duke were to confine it's cuts to research itself, it would have to make massive immediate layoffs.
Remember, many of the costs we are talking about here are inflexible, since the buildings that house research have largely been built with financing and need to still get heat, light, and other services, regardless, and those costs aren't going away. Therefore, most of the flexible costs in research are salaries. But massive layoffs would have a huge, and in some cases even catastrophic, effect on the research. I personally think the spread-the-pain approach Duke is taking is far preferable, even if it means that perhaps Duke isn't willing to spend a few extra hundred thousand dollars on a baseball coach just now.
But no, the federal government doesn't fund baseball, even indirectly, at Duke or at any other large research institution.