There does appear to be a potential sweet spot around 72-25 hours before departure, where upgrading with points can sometimes be processed without the additional fuel surcharges being added. I've certainly benefitted from it a couple of times, so it is worth trying.

From a few reports I've seen recently, they may well have improved their system and now charge the full charges, so it might not work out. Worth asking for a cost on the telephone, should award inventory be available.

There is also the option to bid for an upgrade via the online system. That will tell you exactly how much you will be charged if you are successful. Bids usually clear a few days before the flight.

Finally, you can also try at the airport at check-in. Pricing is a bit more dynamic, so it is a case of seeing what is being offered if anything. It will be more likely to be cash only at a US airport, as many agents can't / won't process points.