The OP simply asked if it was possible to pay cash for an upgrade (yes, certainly) and what the approximate price would be (which is more difficult to say, but worst case, the face difference between what the OP paid and the lowest walk-up price for a business-class ticket, less change fees).

"Total waste", without further elaboration, seems a very strong and subjective evaluation.

Value is often in the eye of the beholder, and while I agree there does not seem to be much tangible advantage in upgrading a short-haul flight on LH, much depends on the actual price of the upgrade, and the OPs value perception. There are some small benefits:

1.) a seat at the front of the airplane, with priority boarding,
2.) a guaranteed empty neighbor seat,
3.) perhaps more attentive and personal on-board service,
4.) perhaps a printed menu listing the offering of the caterer, slightly improved food offerings, and definitely improved drink offerings.
5.) drinks served in glass/ceramic glasses/mugs.

Combined with a smug, patriciate feeling of flying in "business class", I would not be willing impart my value system onto someone else who is willing to pay extra for such benefits. @:-)