As far as the skip option there are two ways to do this the first (possibly easier to implement) is to make it so that the hitching ring in the MC's will either skip to the end of the sentence or reduce the sentence timer by a minute, simple but admittedly a tad boring but is easy and can be used in the mean time as a place holder until the prison is fleshed out more
The next option could be to add a new conversation topic to the Guard where the MC asks her (either innocently or with bratty snark) to "tuck you in" where the guard would proceed to either tie you down to a bed or possibly into a sleep sack with a few toys to keep you entertained (read edged and denied) until your sentence has ended possibly with some internal commentary from the MC about "this not being what they had in mind" or from the Guard about how much fun they are going to have teasing the MC for the rest of their sentence as the screen fades to black to skip some time ahead (personally I say add one or two days to show that time has passed in game). Additionally after the MC has been "tucked in" you could do a follow up scene as they are being released where the MC begs the guard to let them cum as they have been teased and denied the whole time, this could lead to a few options such as the guard refusing the MC as criminals shouldn't be rewarded for their crimes or the Guard could be willing to let them cum on the condition that the MC lets the Guard do something to them either adding a bondage item to them (chastity belt or perhaps), maybe even having the MC eat them out, or have the MC agree to wear something to remind them of how naughty they have been (embarrassing body writing or a collar with something humiliating on it) however after the Guard has had their fun with the MC they could just decide to not keep their word and kick the MC out without any release. This is just an example of one scene that could skip to the end of a prison sentence but basically these scenes could act as a means to end a prison visit as fast or as slowly as the player wants as instead of using another timer the player could explore the prison then start a punishment that would then end the sentence once it was over. You could also use the existing scenes that are already in the prison now but instead of either struggling out of whatever contraption the MC was tied to you would skip to the end of the prison sentence after the interaction ends (maybe add in some more dialog to each scene to compensate), this would also allow you to pad out the prison and prevent players from doing everything that the prison has to offer in one visit then getting board with nothing to do and I guarantee even the masochists who like the idea waiting around twiddling their thumbs and doing nothing will eventually get board as well once they have done everything there after being sent their 100 times as well.
Given this option would require some work/writing/coding, so I would suggest implementing a bare bones time skip first, but having scenes like the one above where a single event could be played out that covers the entire prison sentence would be much better as it would give players a reason to not only to want to go to prison but allow you to control how fast the prison's storyline progresses during each visit while still making each visit worth the time.
Alternatively you could also add an option into the games options menu to adjust the speed at which the timers (prison, day/night cycle, notoriety, ect) in game progress.