I've set up a simple appJar UI, which has an "execute" button that calls a function containing code that takes a minute to run. I have injected my gui() variable, app, into this function.
There are 4 major steps, after each of which I would like a Statusbar to update to reflect that a step has been completed. However, what tends to happen is that as the function code runs, the GUI becomes unresponsive and it isn't until the code completes execution that ALL of the changes to the status bar are displayed at once.
My question is how should I be handling the UI such that the Statusbar is updated in real time?
appJar is just a wrapper around python tkinter module from standard library.
While your code is running, the ui is not running, thus it becomes unresponsive. If you want the ui to remain responsible, you have to return control from your code to the ui library from time to time. 
That can be done by calling gui.topLevel.update() in your code, or by using asynchronous programming and having the main async loop call it, or by using threads.
Which one of those is the best, depends on what your program is doing.
appJar status bar doesn't update in real time