The sun would rise and set later than usual in the spring and fall after the equinoxes, but sooner than usual in the summer and winter after the solstices. The reason for this is simple physics: Summer has longer days while winter has shorter days.
This is because solar radiation increases as a function of the number two solar flares activity in the sun. When there are more sunspots, more radiation reaches Earth's atmosphere. More radiation means longer days during the summer and shorter days during the winter.
Longer nights in the summer and shorter days in the winter with no other changes in climate are called "solstice seasons".
During a solstice season, the length of day and night at the poles is nearly the same as it is near the equator. At the northern and southern poles, daytime reaches its maximum and minimum temperatures, respectively. At mid-latitudes, daytime temperatures are usually very close to their maximum values; at tropical latitudes, nighttime temperatures often reach their lowest value of the year.
Daylight savings time is used by many people all over the world when there is much of a difference between daylight hours during summer and winter months.
Because our globe is tilted on its axis, the sun's path through the sky changes throughout the year. Days are getting longer as we move into spring, so there's no reason for the sun to set any later than it does now.
The earth's axial tilt affects when the days are longest and shortest. It has nothing to do with the season; it happens all year round. But since the rotation of the earth takes place over a period of 24 hours, the time it takes for the rotations to repeat themselves (the "day") varies depending on the position of the earth relative to the sun. Near the vernal equinox, day and night are of equal length. As we move away from this point, the day gets longer and longer until at the poles it reaches 12 months. Then it starts getting shorter again until it becomes six months long at the equinoctial line where day and night are once more of equal length. In other words, the length of day varies between these two values, which is why plants and animals have to deal with both short days and long nights.
The amount by which the earth's axis is tilted determines how much of a difference there is between day and night throughout the year.
The rising and setting times vary slightly from day to day. At the summer solstice, the sun rises as far northeast as it ever does and sets as far northwest as it ever does. The sun rises a little more south every day after that. The sun rises straight east and sets due west during the autumn equinox. It's highest in the sky at midday and lowest at midnight during the winter solstice. The spring equinox marks when the sun is exactly over the equator, so it's not higher or lower in the sky than at any other time of year. The fall equinox marks when the sun is exactly opposite the equator, so it's not lower or higher in the sky than at any other time of year.
At geographical north, the angle between the horizon and the vertical is called the nadir. Nadir points change throughout the day; it's directly overhead at noon but near the ground at midnight. Rising and setting stars are visible for several days in both directions from the equator. They're always north of the celestial north pole, which is fixed against the rotating earth.
Southern hemisphere observers experience their own unique astronomy during the night of June 20th. The moon is on the other side of the earth, so it appears as ILLUMINATED by the sun during part of the day. This is called "moonlight" and it's when most lunar eclipses occur.
The sun rises sooner and sets later in the summer, thus there are more hours of daylight. When clocks and other timepieces are set ahead by a certain amount (typically one hour) in the spring, the sun will rise and set later in the day as recorded by those clocks. Thus, the sunrise is earlier in spring.
The reason for this is that plants need sunlight to grow and flowers need light to bloom. In the winter, plants go into hibernation and don't use energy so they don't grow or flower. In the spring when temperatures increase, plants start growing again and use energy so they make more food via photosynthesis. This takes energy which can be used for other things than just living - for example, making seeds or building branches. The more energy plants use, the later it gets up in the morning and uses up all its stored energy at sunset. In the fall, when the temperatures drop, plants lose their growth habituation and start making food again.
This is why you usually see more flowers and plants around during the springtime - because they're making more food using up all their stored energy from the previous season. Without these reserves, they wouldn't be able to survive through such a difficult time of year. Winter comes once per year but spring comes every month on Earth!
And here's another thing about plants - some need only half of the sunlight that others do.
The condition you describe occurs when the winter solstice approaches. In contrast, as we near the summer solstice, sunlight becomes earlier and dusk becomes later. This is because the Earth circles the Sun at an irregular pace. When it is closest to the Sun, on January 3rd, it takes 24 hours 48 minutes for a circle to complete itself. At its farthest from the Sun, on July 4th, it takes 24 hours 56 minutes for a circle to complete itself.
This effect is called "seasonal lag" and it causes changes that occur in one season to be delayed relative to similar events in another season. For example, if the winter solstice were today, night would still be falling when the Romans celebrated their New Year in December. The reason is that this year the spring equinox occurred on March 20th, which means that today is actually the first day of spring!
Seasonal lag causes differences between the seasons that appear wrong or unnatural. For example, during the winter months in the Northern Hemisphere, daytime temperatures are low while nighttime temperatures are high. The opposite is true in the Southern Hemisphere during the same time period. The cause of this phenomenon is simple: seasonal lag results in the days getting longer as we approach the spring equinox and shorter during the fall back toward the winter solstice.
Solar noon happens a few minutes later than the previous day around the solstices. Sunrises and sunsets are getting later and later each day following the winter solstice, as solar noons get later and later. This is why a location's earliest sunset occurs before the winter solstice and its latest dawn happens after. During the middle of the year, when days are longest, solar noon occurs near midday; during the winter, when nights are longer, solar noon comes later in the afternoon or even evening.
--from "What time is sunrise on the equinoxes?"
Sunrise is when an arc of sunlight reaches the horizon, regardless of the season. Therefore, the earliest sunset before the summer solstice and the latest sunrise after the winter solstice. Sunset is when an arc of sunlight disappears below the horizon. Thus, the earliest sunset occurs before the summer solstice and the latest sunrise after the winter solstice.
--from "When does the first star appear in the night sky?"
The first star appears just before midnight at the beginning of spring. The reason that this happens before midnight is because daytime begins to lengthen at mid-northern latitudes like here in North America when it's springtime.
--from "How do I know if it's going to rain? How do I find out the weather forecast?"