Reaganomics has more to do with Hayek's ideas than with those of Keynes.
Reaganomics was a supply-side approach to economics. Reagan wanted to reduce taxes on the idea that this would increase the aggregate supply in the economy. He felt that reducing taxes would stimulate people and businesses to produce more. He also wanted to reduce government regulations and rely more heavily on monetary policy than on fiscal policy.
This is not in line with Keynesian ideas. For Keynes, fiscal policy was meant as a way to stimulate aggregate demand, not supply. Reagan's tax cuts were aimed more at the rich (to stimulate supply) than at the middle class (to stimulate demand).
Reagan's ideas were more closely in line with those of Hayek since they both wanted less government involvement in the economy and were also more inclined to use monetary policy than Keynes was.