Does Earth move faster in its orbit near perihelion (January) or near aphelion (July)? Based on your answer, is the solar day longest in January or July?
According the Kepler's laws of planetary orbits, a planet is moving at its fastest at perihelion. It is moving at its slowest at aphelion.
Currently the Earth is at perihelion around 3 January and at aphelion around 3 July. The actual time of perihelion varies from year to year by a few days - usually between 2 and 4 January. The reason for this being that the Earth's orbit is constantly changing due to the gravitational effect of the other planets.
Also, perihelion is getting progressively later due to precession. It gets later by about a day every 70 years.
A solar day is the time between two successive solar noons - the time when the Sun is at its highest in the sky. The difference between solar noon and clock noon is defined by the equation of time. This has two components, one due to the Earth's orbit being an ellipse, the other due to the Earth's axial tilt.
The diagram shows the equation of time. The solar day is currently at its longest in December around the time of the solstice.