It actually is possible, especially as to a green hand like Ishmael, which is perhaps unsurprising as Melville based several of his novels, including Moby Dick, on his life experience as a green hand in the whalefleet.[1]

A whaling ship[2] is a significant capital investment. A whaling ship could be owned by a single wealthy merchant, or owned by a group, legally a partnership, of successful merchants. The number of partners would vary, from two to a half-dozen or so. Sometimes a successful whaling captain would join a partnership and then captain a vessel that he (part-) owned, but usually the owner or consortium of owners would hire a captain that they trusted, and then the rest of the crew would be hired by an agent that worked for the ship owner(s). Experienced crew would want to know who the captain was before they signed on, as a captain's reputation proceeds him. Even if the captain was on his first voyage as captain, he already had a reputation from his time as a mate. Whaling captains and whaling crew all lived in the same communities, so any experienced whaler would have met his future captain in church, in the town militia, in the shops and stores, and so forth. But Ishmael is specifically from out of town, and nearly immediately signs on to a ship once he gets in Nantucket, so it's quite understandable that he hadn't met the captain socially prior to the voyage. His friend the "harpooner"[3] Queequeg, though, would likely have heard something about every captain in the region, including Ahab.

While the captain could take personal charge of the vessel when leaving port, he was under no obligation to do so; that sort of routine seamanship was well within the capabilities of the mates.

And the scene in the book when Ahab comes out and gives a big speech to the crew, when he declares that they're going to hunt the white whale, and there's cheering and whatnot? That was accurate too, at least the speech part! It was tradition that once the ship was out of sight of land, all hands would be called the the quarterdeck, and the captain would deliver a "Harangue". (The word didn't have a negative connotation in the 1800s.) The captain would address the crew and tell them the general plan for the voyage, including expected stops on the way and where they hoped to catch whales, and he would encourage them to work hard with tales of the riches they would gain.[4] My impression is that it served the same purpose as a locker-room speech-- the captain is trying to bond the crew together, create loyalty to the officers, and inspire everyone to perform well.

So is it realistic? No, in that most crew members would have met the captain in ordinary social situations in their home port. But for a green hand, new in town, it's quite possible he would first lay eyes on the captain when the ship was a dozen miles offshore and the captain called the crew together to deliver his harangue.

[1] Technically Melville sailed on three different whaling ships- he signed on to the Acushnet in New Bedford, "jumped ship" in the South Pacific, but was eventually picked up by the Lucy Ann. He never officially signed on to the second ship but one assumes they made him work for his passage, which ended with him in jail. Once out of jail, he signed on the Charles & Henry for his third and final experience of whaling.

[2] I'll call all whaling vessels "ships", but at the time they were distinguished by a more specific name that referred to their sail plan (the arraingement of masts and sails)-- so one might sail on a whaling sloop, a whaling brig, or a whaling ship, but anyone that called a whaling sloop a whaling ship was a landlubber not worthy of attention.

[3] American whalers didn't use "harpooner" as a title. The position was actually called "boatsteerer" as they would be directing one of the small whaleboats that where launched from the whaling ship to do the actual harpooning. They were not officers, though, in that they didn't command the larger vessel and were not trained in the vital officers' skill of celestial navigation. I suspect Melville used "harpooner" because it's a much more exciting word than "boatsteerer", but that's a literary analysis question, not a historical one.

[4] Whaling crew were unusual in that they were not paid a salary. Instead every sailor, from the captain down, contracted for a "lay" which was a portion of the voyage's net income revenue (not profit!), with the only expense to be subtracted that of shipping home oil, which would only happen if the ship was so successful its own hold was full. Looking at an example before me (the 1873 voyage of the bark Matilda Sears), the captain had a lay of 1/16, four "ordinary seaman" got 1/185 each, the cook got 1/160, the ship's boy received 1/300 etc. There were twenty-six crew in total, and collectively the entire crews' lays sum to 30.15% of the revenue, with the rest going to the ship's owners, out of which they'd have to pay for all the supplies and repairs.

EDIT: Removed a few typos & expanded slightly on the lays.