When a high pressure system moves over an area, the presence of descending air instead of ascending air provides stable atmospheric conditions that are <em>not favourable </em>for the formation of clouds. Because of this, high pressure systems are not associated with rainy weather.
We can't really make a prediction about exact temperature without more information, but because high pressure systems feature clearer skies, it is possible that the days are warmer and the nights are cooler than before the passage of the system. There are, of course, exceptions and in the winter a dome of frigid high pressure can keep even daytime temperatures very low.
