Ok, so if the pulley circumference is 15.7, and the slider is 60cm, and the stepper is a 48-pole stepper, then that comes to about 183 steps.
Oh, and this is quite slow - a time-lapse situation. So you'll set the thing up initially by hand, dial in the time, and then hit the button. No need for a UI to move the stepper back and forth, to reset its position.
And you want to use a potentiomter? A potentiometer on an analog port has a resolution of 1/1024. If the upper bound for your timing is hours, then one "tick" of the pot will be three seconds … and even that's not realistic to expect because we are talking very fine adjustment.
Another wrinkle is that the clock on an arduino UNO is not terribly accurate - it can be off by as much as 10%.
But … if it doesn't really need to be as accurate as all that, if ballpark is acceptable, then this is do-able. Set time with pot, display on LCD, then when the button is hit begin stepping and display amount remaining. Power cycle the whole dooverlackie to reset.
The main difficulty with programming this is setting up all the hardware. I have all the bits you are using, but it's all in boxes. The other difficulty will be getting all the pins and whatnot on my setup to match your setup.
Then again … I've been looking for an LCD project, just to get my feet wet.
Look - it's laundry day, and my 3d-printer has been disappointing me for days. I'llput it away and have a bash at this, but I will have to ask for more than my usual fifty buck minimum for an easy project because of all the hardware setup. I'll say $100 AUD, and that's for the very basic thing I have outlined here. Payment on delivery, because there's never a guarantee that the project won't blow out.