Did you design your GUI using guide? That will make things much easier, especially if you're new to Matlab as you mention.
If the GUI is made in guide, an mfile will be generated once you save the GUI. After that, you would just need to put your code in the pushbutton's callback. I wouldn't think it would take a ton of code to do what you want, just something like this:
function pushbutton1_Callback(hObject, eventdata, handles) %button callback
func_name = get(handles.editbox1,'String'); %name you entered in edit box
myfun = str2func(func_name); %turn the string into a function handle
myfun_output = myfun;
%some formatting of myfun_output here perhaps, for readability and to
% convert it to a string.
set(handles.listbox1,'String',myfun_output) %your listbox
guidata(hObject,handles)
Your main question, though, just seemed to be about being able to call the function from within your GUI's mfile. To do that, all you should have to do is call the function from within the button's callback. Depending on the needed inputs to your function, things could get a little more complicated, but the calling one function from another and saving the output part is relatively simple.
