Oh, Putin won't attack Falkland directly but he can sponsor invasion by Argentina who indeed have a claim to this land. UK will not go into all out war with Argentina, they will just sanction them and refuse to do business with em, exactly what Russia wants in a way. All Argentina have to do is send 2,000 troops and 20 tanks against 40 UK troops and UK troops will surrender. They will take over the land with no shots fired... Just like in Crimea. Putin invaded Crimea with 3,000 soldiers against Ukrainian military that had 120 troops... 120 troops surrendered and were sent home. No one got hurt.
Putin then can sign a lease agreement with Argentina and make Falklands his for a small fee.
The UK would absolutely fight to keep the Falklands. Any UK government that allowed them to fall would face a vote of no confidence in a matter of days. And UK defenses are far more robust than they were in 1982. A full strength regiment of well trained and equipped British infantry, along with a squad of Typhoon fighters, and SAMS. They can be reinforced quickly, and the Royal Navy can have nuclear subs there in a matter of days.
The most advanced fighter in the Argentinian AF is basically the same fighter they used in 1982. The British would be able to control the air and sea around the Falkland's. With no land border, you can't win a war if you can't control the air and sea.