For a given date, longitude, and latitude:
Popular apps and websites give the time when the sun's center or edge is on the horizon (i.e., the time of sunrise and sunset) or its transit (i.e., noontime), but I want to ask how to calculate the time needed for the solar disk to pass the horizon's line and the transit's line (preferably in c# or java)?
The time it takes the sun to cross the horizon is given at:
Re the time it takes to cross the transit line, this doesn't really answer your question, but, to a good approximation, the time varies between 128 seconds at the equinoxes and 140 seconds at the solstices, regardless of latitude or longitude.
More specifically, the formula is $\frac{128}{\cos (\text{dec})}$ seconds, where $\text{dec}$ is the Sun's declination. You can calculate the Sun's declination itself using the formulas at https://en.wikipedia.org/wiki/Position_of_the_Sun
The calculation here is relatively simple:
The Sun travels $360 \cos (\text{dec})$ degrees in a 24-hour day, where $\text{dec}$ is the Sun's declination
When the Sun is transiting, the motion is perpendicular to the transit line (the Sun's motion is entirely in azimuth, not in altitude)
Therefore, all of the Sun's angular motion translates to motion across the transit line; in contrast, the Sun rises and sets a (non-perpendicular) angle (except at tropical latitudes on the two days where the Sun passes directly overhead), so sunsets and sunrises take longer than $\frac{128}{\cos (\text{dec})}$ seconds
Since the Sun has an angular diameter of 32 minutes or $\frac{8}{15}$ degrees, it takes $\frac{\frac{8}{15}}{360 \cos (\text{dec})}$ of a day for the Sun to cross the transit line
Since a day is 86400 seconds, this works out to $\frac{128}{\cos (\text{dec})}$ seconds.
Caveats and nitpicks:
I assume there are 86400 seconds between successive noons. This is incorrect for two reasons:
The time between noons is not exactly one day. The cumulative difference forms the Equation of Time but the day to day difference is quite small
The Earth's day is slightly longer than 86400 seconds, which is why we need leap seconds
The Sun's angular diameter actually varies based on Earth's distance from the Sun, but 32 minutes is a good approximation
I assume the Sun's altitude doesn't change while it's transiting. While this is a good approximation, the altitude does change slightly
I assume the Sun's declination doesn't change while it's transiting. The change in declination is very small, so this is a reasonable assumption
There are probably other assumptions I made implicitly that I am not noting here.
Refraction is not an issue, since the motion we are discussing is azimuthal and not in altitude.
I used the "one over cosine" form above to make things easier for non-mathematicians. The more compact form would use "secant".
The calculations I did for this problem are disorganized, but available at: https://github.com/barrycarter/bcapps/blob/master/STACK/bc-solar-transit.m
