Not really that easy. You can fly from Punta Arenas every Saturday. So, bus from Ushuaia, but then you would be there for a week minimum.

There is also a stop of that flight in Rio Gallegos (ARG) once a month (or used to be). But you would still have to stay the week.

(There are 2 direct flights to the UK per week from the Falklands, but these are expensive. But it would be possible to arrive from South America on a Saturday and leave on a Tuesday. However, for a variety of reasons, I would not recommend such a short stay.)

The only other alternative might be a cruise or expedition ship going to Buenos Aires or South Georgia or Antarctica, which calls in to the Falklands.

Lastly, you need to be aware that there are political issues which mean the relationship between Argentina and the Falklands is not that great. Some Argentine politicians claim the Falklands as part of Argentina.