The public access to the property or the method means that if you start your app from the outside, from the Command Window like this, you will be able to get property named "Property" as defined in that block.
So, what you need is to reference the property of the app correctly.
properties (Access = public)
Notice, that your app has now property A, and you index into it like this
So, you can assign a value to property like this
function StartButtonPushed(app, event)
app.A = app.Spinner1.Value;
That should be it, before asking for similar questions, take a look at tutorials here and introductory apps in the App Designer, and there are plenty of questions in MATLAB Answers.