How can I find out if the government paid to build a runway?
The FAA gives funds to airports through the Airport Improvement Program (AIP) and that could include runway improvements. It's public money so the information is also public and should be in the Excel files on that page. If that isn't what you need, you can always submit an FOIA request to the FAA for all information they have about the airport.

It's also possible that you could get more information from the state or county. They usually have at least some input into airport construction, because of local zoning laws, easements and so on. Even if the rumor is false and the runway is built on private land and paid for with private money, there may be county planning or landowner records. But that's definitely beyond the scope of this site.