It was quite common for the Royal Navy to send ships on solo missions during the Napoleonic period (which Hornblower and "Master and Commander" are set during), but it would be quite uncommon to send a first- or third-rate ship on such a mission.

Most of the types of ships that would be sent on single-ship missions were of the scouting or messaging variety; in the days before telegraph or radio, sending quick-sailing ships between fleets and ports was the only way to convey information. It was common, for example, for a single small ship to carry dispatches back to England after a fleet action or battle; being picked for such was considered a mark of honor for the dispatch ship's captain.

Frigates were often used to cruise, either singly or in groups, against enemy shipping. In 1795, two independent squadrons of frigates under Edward Pellew and Sir John Warren were based from Falmouth, where they operated against the French coastal trade. N.A.M. Rodger in "Command of the Ocean" says they reduced Brest to "near starvation" in 1795, and relates an anecdote in which Sir Sidney Smith took HMS Diamond "into the Goulet by night in January 1795, hailing the French ships in his faultless French to ask for news, and returning without detection with the latest information."

Scouting and intelligence work was a very common mission for frigates. Shortly after the outbreak of the French Revolution, in 1796, the Channel Fleet received intelligence that the French fleet in Brest was preparing to embark troops for an unknown destination (which proved to be Ireland); at the time the Channel Fleet moved to cruise to the west of Ushant and sent Pellew alone into the Iroise to watch the harbor at Brest, for example. (In that instance the British couldn't prevent the French squadron from slipping out, due to a combination of bad weather and worse luck.) Pellew's mission would be repeated throughout the war as the main British fleet cruised off Brest and Toulon and fast frigates were used inshore to watch the French fleets and their preparations.

When the Spanish revolt broke out in 1808, several frigates based in Port Mahon raided coastal shipping and cruised in support of the Spanish rebels in Catalonia, most notably Lord Cochrane (whose career the fictional Jack Aubrey parallels) who captured a French fort with his ship's boats and engaged in other feats of daring.

During the war of 1812, several single-ship actions were fought between British and American frigates, mostly to the Americans' advantage; but the best-known British victory of the time came when HMS Shannon was watching Boston alone and USS Chesapeake stood out to fight her. (Chesapeake was boarded and captured after a short action.)

To turn to your question more directly (and the events depicted in Master and Commander), yes, ships were sent on independent cruises halfway or more around the world. Frigates were often used for this because they were both large enough to be a credible fighting force and fast enough to cover the distance in a a reasonable amount of time, but other vessels also set off on independent cruises.

The most famous anecdote (and highly exaggerated) from Horatio Nelson's early career was when he saw and fought a polar bear while on a mission to seek out the Northwest Passage; Nelson was a midshipman on board HMS Carcass, which reached within 10 degrees of the North Pole before being forced to turn back. Later in Nelson's career, he commanded frigates independently in the Caribbean and the Mediterranean, and as a fleet admiral employed frigates to scout and send messages throughout the Mediterranean.

Now, to your question related to whether a first- or third-rate would be sent on such a mission: Most likely, no. Those ships were far too expensive, and slow, to be used in most missions that frigates and smaller ships could be used for, and could on occasion be vulnerable to a squadron of smaller ships. Most famously, Edward Pellew in HMS Imperiuse, in company with HMS Amazon, fought a running action with the French third-rate Droits de l'Homme that resulted in the French ship's destruction in 1797. Amazon was also lost in the action as Amazon and Droits de l'Homme were both driven ashore, but the result was a net loss to the French.

Sources I used directly for this: N.A.M. Rodgers, The Command of the Ocean; John Sugden, Nelson: A Dream of Glory; Thomas Cochrane, The Autobiography of a Seaman. There are plenty of others out there.

edit: formatting