A paid (but reasonably priced, US$49.95) program known as SnagIt for Windows and Mac may serve your purposes. It's a rather capable screen capture utility that allows the user to select entire screen, specific windows, selected areas (drag out a box) and perform various actions on the captured information.
For your question, and according to this tutorial, SnagIt will do exactly as you require. My experience with the software also confirms this capability.
You would use the properties option for your capture to set a specific window size and location as well as the actions to be performed (Grab Text) to create a profile within the program. This would then be activated each time you press the Print Screen button (or any user selected keys).
SnagIt also allows automatic file saving, although I did not research the OCR aspect of this feature. With "ordinary" graphics captures, the user specifies a folder destination, a file name prefix and an increment figure. An example of the default filename prefix is snagit followed by 0000 and the appropriate extension (.jpg, .pdf, .txt, etc.). Each press of the hot key then increments the counter and stores the information.
I typically disable the preview feature in properties, as I usually don't need to see each captured screen until later. If it's a complex profile, I might leave preview enabled for one or two captures.
The publishers of the program offer a free trial, which I suspect would be valuable if you should discover that the features do not meet your requirements.